Accretion driven turbulence in filaments I: Non-gravitational accretion

TL;DR

This study investigates how accretion-driven turbulence influences star-forming filaments, revealing a linear relation between velocity dispersion and inflow velocity, with turbulence providing additional support in filament stability.

Contribution

It demonstrates that non-gravitational accretion alone can generate turbulence consistent with observations, highlighting its role in filament structure and stability.

Findings

Velocity dispersion linearly related to inflow velocity

Turbulent velocity dispersion is independent of sound speed

Non-thermal line widths are generally subsonic

Abstract

We study accretion driven turbulence for different inflow velocities in star forming filaments using the code ramses. Filaments are rarely isolated objects and their gravitational potential will lead to radially dominated accretion. In the non-gravitational case, accretion by itself can already provoke non-isotropic, radially dominated turbulent motions responsible for the complex structure and non-thermal line widths observed in filaments. We find that there is a direct linear relation between the absolute value of the total density weighted velocity dispersion and the infall velocity. The turbulent velocity dispersion in the filaments is independent of sound speed or any net flow along the filament. We show that the density weighted velocity dispersion acts as an additional pressure term supporting the filament in hydrostatic equilibrium. Comparing to observations, we find that the projected non-thermal line width variation is generally subsonic independent of inflow velocity.

Figures

Equations

$$\rho (r)=\frac{{\rho }_c}{{\left(1+{\left(r/H\right)}^2\right)}^2}$$

$$H^2=\frac{2c_s^2}{\pi G{\rho }_c}$$

$${\left(\frac{M}{L}\right)}_{\text{crit}}=\frac{2c_s^2}{G}\approx 1.06\cdot 10^{16}{\text{g cm}}^{-1}\approx 16.4{\text{ M}}_{\text{sun}}{\text{ pc}}^{-1}$$

$$a=-\frac{2GM/L}{r}$$

$$E_{\text{pot}}=2G(M/L)m\ln \left(\frac{R_0}{R}\right)$$

$$v_r=2\sqrt{G(M/L)\ln \left(\frac{R_0}{R}\right)}$$

$$\dot{M}={\rho }_0{v}_{r}2\pi R_{0}L$$

$$\rho (r)={\rho }_0\frac{R_0}{r}$$

$$M(R)={\rho }_0{\mathcal{M}}^2\pi R{R}_{0}L$$

$$R(t)=\frac{2c_s^{2}t}{v_r}$$

$${\tau }_d\approx \frac{L_d}{\sigma }$$

$${\dot{E}}_t={\dot{E}}_a-{\dot{E}}_d=(1-ϵ){\dot{E}}_a$$

$${\dot{E}}_a=\frac{1}{2}\dot{M}{v}_r^2$$

$${\dot{E}}_d\approx \frac{E_t}{{\tau }_d}=\frac{1}{2}\frac{M{\sigma }^3}{L_d}.$$

$$ϵ=\left|\frac{{\dot{E}}_d}{{\dot{E}}_a}\right|$$

$$\sigma ={\left(2ϵR(t)v_r^2\frac{\dot{M}}{M(t)}\right)}^{1/3}$$

$$\sigma \sim v_r^{1/3}.$$

$${\sigma }_s^2=\ln \left(1+b^2{\mathcal{M}}^2\right).$$

$$\sigma =0.30v_r-0.60c_s.$$

$$P_{\mathrm{turb}}=⟨\rho ⟩{\sigma }^2,$$

$$P_{\mathrm{ram}}=⟨\rho ⟩{⟨v_r⟩}^2$$

$${\sigma }_{\mathrm{NT}}=\sqrt{\frac{{\text{FWHM}}^2}{8\ln 2}-\frac{k_{B}T}{m}}$$

$$\sigma =0.30v_r-0.60c_s$$

$$\frac{1}{2}\dot{M}{\sigma }^2=0.09\cdot \frac{1}{2}\dot{M}{v}_r^2-0.18\dot{M}{c}_s{v}_r+0.18\dot{M}{c}_s^2$$

$$v_r\ge 2.0c_s.$$

$${\dot{E}}_t=\alpha {\dot{E}}_a-{\dot{E}}_d$$

$$\alpha =\mathrm{0.09.}$$

$${\dot{E}}_d=0.18\dot{M}{c}_s{v}_r-0.18\dot{M}{c}_s^2=0.18\dot{M}\left(c_s{v}_r-c_s^2\right)$$

$$R_i=bR(t)=b\frac{2c_s^{2}t}{v_r}$$

$$M_i=\rho (R)\pi R_i^{2}L={\rho }_0{\mathcal{M}}^2{R}_0\pi b^{2}RL=b^{2}M(R)$$

Full text

††pubyear: 2017††pagerange: Accretion driven turbulence in filaments I: Non-gravitational accretion–Accretion driven turbulence in filaments I: Non-gravitational accretion

1 Introduction

Turbulent motions are ubiquitous on all astrophysical scales. There is evidence for highly complex non-thermal motion from the intergalactic medium to the interstellar medium (ISM) and individual molecular clouds down to even the smallest scales of protostellar discs. Likewise, as part of the ISM, star forming filaments are no exception to this observational fact. In contrast to its importance the origin of turbulence is still not fully understood and there are numerous potential sources for turbulent motions in the ISM (Mac Low & Klessen, 2004; Elmegreen & Scalo, 2004; Elmegreen & Burkert, 2010; Klessen & Glover, 2014). On cloud scale, molecular line observations are dominated by supersonic motions and show a direct correlation between size and line width (Larson, 1981). This is usually interpreted as the direct result of a turbulent cascade from the scale of a tens of parsec sized molecular cloud down to the scale of parsec sized filaments (Kritsuk et al., 2013; Federrath, 2016; Padoan et al., 2016). Filaments then inherit their internal velocity dispersion from the motions on larger scales. This model is also favoured by Herschel observations which show that filamentary structures are ubiquitous in molecular clouds (André et al., 2010; Arzoumanian et al., 2011; Arzoumanian et al., 2013; André et al., 2014). This picture has recently been challenged by the discovery that more massive filaments are actually complex bundles of fibers whose line-of-sight superposition create the observed supersonic linewidths (Hacar et al., 2013) and which were also found to form in numerical simulations (Smith et al., 2014; Moeckel & Burkert, 2015). Independent of the formation process, we argue that accretion driven by the gravitational potential of the filaments alone can be enough to stimulate turbulent motions that are in agreement with the observations (see also Ibáñez-Mejía et al., 2016).

Recent images of the high column density gas, traced by the usually optically thin  C18O line of nearby filaments have shown that the non-thermal linewidth is predominantly sub- and transsonic along filaments. This is true for L1517 in Taurus (Hacar & Tafalla, 2011), where the mean is about half the sound speed, as well as for the fibre-like substructure of the L1495/B213 region in Taurus (Hacar et al., 2013), where the mean is about the sound speed. Moreover, even in the Musca filament, a 6 pc long structure, subsonic non-thermal linewidths dominate along the filament (Hacar et al., 2016). Larson was the first to connect the velocity dispersion to the size of molecular clouds (Larson, 1981). While not strictly valid for filamentary structures, it predicts a factor of about three times higher velocity dispersion corresponding to the supersonic regime for a filament like Musca with a width of 0.14 pc (Cox et al., 2016). Although still within the spread of the overall relation, Musca is definitely an outlier in terms of its length and it is unclear which process leads to a structure of that size and that low velocity dispersion. In this paper we explore a possible origin of turbulence in filaments.

In the following sections, we introduce the basic concepts we use to constrain our model (section 2). We then discuss the code and the numerical set-up (section 3). Thereafter, we present our results of the simulations and discuss them in detail (section 4). Additionally, we show that turbulence plays a role in creating a hydrostatic equilibrium (section 5). Finally, we compare our data to the observations (section 6) and investigate the dependence on filament inclination.

2 Basic concepts

In order to sustain turbulence inside a filament there has to be an external driving mechanism. Otherwise, turbulent motions decay on the timescale of a crossing time (Mac Low et al., 1998; Stone et al., 1998; Padoan & Nordlund, 1999; Mac Low, 1999; Mac Low & Klessen, 2004). Here, we discuss a possible source of the external driving and the theoretical prediction.

2.1 Gravitational accretion onto a filament

Although there are different ways to accrete mass onto a filament, e.g. a converging flow, these processes are typically limited in time. A counterexample for a radial converging flow which is stable over longer timescales is the gravitational attraction of the filament itself. If we assume that the filament is isothermal and in hydrostatic equilibrium, then it has a density profile first described by Stodólkiewicz (1963) and Ostriker (1964):

[TABLE]

where $r$ is the cylindrical radius and $\rho_{c}$ is its central density. The radial scale height $H$ is given by

[TABLE]

where $c_{s}$ is the isothermal sound speed and $G$ the gravitational constant. We assume that the gas has an isothermal temperature of $10\text{ K}$. Using a molecular weight of $\mu=2.36$ gives the isothermal sound speed of $c_{s}=0.19$  km s*-1*. One can integrate the profile to $r\rightarrow\infty$ to get the critical line mass of

[TABLE]

above which a filament will collapse under its self-gravity. Following Heitsch et al. (2009), for a given line mass $M/L$ the gravitational acceleration of the filament is:

[TABLE]

One can calculate the potential energy which a gas parcel of mass $m$ loses in free-fall starting with zero velocity at a distance $R_{0}$ to the filament radius $R$ by integrating over $r$:

[TABLE]

Therefore, the inflow velocity at the point of accretion $R$ is:

[TABLE]

Note that similar to the free fall velocity, the value does not depend on the mass of the gas parcel. It is also not sensitive to neither the starting position nor the line mass while depending stronger on the latter. As the filament accretes mass and increases in line-mass, the inflow velocity grows. However, as we want to analyse accretion driven turbulence in an equilibrium state we keep the inflow velocity constant. Therefore we choose to neglect the effects of gravity and use an artificial but constant mass inflow where we set the inflow velocity also to a constant value. The effects of gravity will be discussed in a subsequent paper. Assuming the extreme case of a filament with a gravitational influence of a hundred times the filament radius, which for a typical radius of 0.05 pc (Arzoumanian et al., 2011) is the size of a typical molecular cloud, we still need a line mass which is several times higher than the critical line-mass to achieve an inflow velocity of even Mach 10.0. Thus, we limit our maximum inflow velocity to Mach 10.0.

A consequence of a constant inflow velocity is a constant mass accretion rate. The absolute value is set by the radius of the inflow region $R_{0}$ and the density at that radius $\rho_{0}$:

[TABLE]

This should stay constant for every radial shell and thus leads to the following density profile outside of the filament:

[TABLE]

As there is an isothermal accretion shock formed at the filament boundary, pressure equilibrium requires that the mean density inside the filament is the outside density times a factor of the Mach number $\mathcal{M}$ squared. This leads to the following filament mass-radius relation:

[TABLE]

This has to be the same as the accreted mass given by the mass accretion rate given by Equation 7 times the time $t$. Therefore the radius of the filament evolves as:

[TABLE]

2.2 Turbulence driven by accretion

Following Klessen & Hennebelle (2010), Heitsch (2013) derived an analytical expression for the velocity dispersion depending on the inflow velocity. We expect turbulence to decay on the timescale of a crossing time:

[TABLE]

where $\sigma$ is the velocity dispersion in three dimensions and $L_{d}$ is the driving scale of the system. Klessen & Hennebelle (2010) use the approach that the change in turbulent kinetic energy is given by the balance of the accretion of kinetic energy and the dissipation of turbulent energy:

[TABLE]

with the energy accretion rate

[TABLE]

and the loss by dissipation as

[TABLE]

They also introduce an efficiency factor $\epsilon$ as fraction of accreted energy which can sustain the turbulent motions:

[TABLE]

Thus, if the driving scale is the filament diameter $L_{d}=2R$, Heitsch (2013) predicted a turbulent velocity dispersion of

[TABLE]

This assumes that $\epsilon$ is independent of the inflow velocity. In our simple case the radius depends on the constant inflow velocity as $1/v_{r}$ and the radial accretion leads to a radius and mass growing linear in time. Therefore, we expect a constant level of velocity dispersion which should behave as

[TABLE]

This is the relationship we want to confirm or disprove using numerical methods.

3 Numerical set-up

We executed the numerical simulations with the code ramses (Teyssier, 2002). The code uses a second-order Godunov scheme to solve the conservative form of the discretised Euler equations on an cartesian grid. For the simulations we applied the MUSCL scheme (Monotonic Upstream-Centred Scheme for Conservation Laws, van Leer (1977)) together with the HLLC-Solver (Toro et al., 1994) and the multidimensional MC slope limiter (van Leer, 1979).

We simulate a converging radial flow onto a non-gravitating filament in order to study the generated turbulence. We use a 3D box with periodic boundary conditions in the x-direction and outflow boundaries in the other directions. The periodic boundary prohibits the loss of turbulent motions in x-direction. As ramses cannot use a radial inflow boundary we define a cylindrical inflow zone which lies at the edges of the box and has a thickness of two cells from where material flows onto the central x-axis of the box. The initial gas density inside of the box is set to a mean of $3.92\cdot 10^{-21}$ g cm*-3*, corresponding to about $10^{2}$ particles per cubic centimeters for a molecular weight of $\mu=2.36$. Additionally, a random perturbation is added inside of the inflow zone at the beginning of the simulation. This is illustrated in Figure 1 where we show a density slice through the y-z plane. The inflow has a constant density of $3.92\cdot 10^{-21}$ g cm*-3*and a constant velocity. Thus, it leads to a build-up of material in form of a filament with a radius that grows over time as it is not restricted by gravity. Consequently, one has to ensure a big enough box that an equilibrium can be established. The surrounding cells around the inflow zone are given a constant density with the same value as the inflow zone and pressure and do not affect the simulation.

The complete box is set to be isothermal with a temperature of 10 K and a molecular weight of $\mu=2.36$. In general, the boxsize is 0.8 pc. For the control runs with a higher temperature the boxsize is doubled to ensure enough space to reach a velocity dispersion equilibrium.

As the inflow initially leads to a thin, compressed central filament with a high density, we employ adaptive mesh refinement (AMR) to resolve high density regions while keeping the resolution low in the remainder of the box. We test different refinement strategies by varying the maximum resolution to fulfill the Truelove criterion for the maximum occurring density (Truelove et al., 1997) while keeping the minimum resolution constant at $256^{3}$ cells. Our initial approach is to resolve the Jeans length by 16 cells. We also test a more conservative criterion suggested by Federrath et al. (2011) using 32 cells to resolve the Jeans length in order to sufficiently resolve the turbulent cascade. However, we cannot detect a quantitative difference in the velocity dispersion to the previous case with lower maximum resolution. Furthermore, even a lower maximum resolution of 8 cells per Jeans length does not show any difference in the value of the velocity dispersion. Despite there being no change in behavior for the maximum resolved density, we see a difference for a varying minimum resolution. We present the details and a resolution study in the next section.

4 Simulations

In this section we present the outcome of our simulations. In order to measure the velocity dispersion and the mass of the filament one has to distinguish between filament material and ambient medium. As the inflowing material is shocked at the filament surface there is a clear increase in density and a clear drop in radial velocity. As the internal density of the filament decreases over time due to Equation 9, we use the radial velocity to distinguish inflowing material from filament material instead of a density threshold. To measure the filament radius we use the mean position of the highest density gradient which traces the general shock position.

4.1 Initial perturbation

As a reference test case we set up a converging flow with a velocity of Mach 5.0 with and without a perturbation of the initial density field. As material streams in, it is compressed on the central axis of the box. In order to avoid unphysical densities in the initial phase of the case without initial perturbation we add an already existing filament to the initial conditions with a radius of 0.05 times the boxlength and with a constant density of ten times the ambient density corresponding to a total mass of $0.23$ M⊙. Note that we do not need to include an initial filament for the simulations including an initial perturbation, as the generated turbulence prevents the density from reaching values which cannot be resolved. For a low resolution there is a stark contrast in the structure between both cases. An initial perturbation leads to a considerable amount of substructure which resembles observed filaments. The case lacking an initial perturbation does not obviously exhibit signs of turbulence being present. The motions are purely angular and radial and the little substructure which is generated is obscured due to the projection. However, as the inflow is perfectly symmetrical, only radial motions should exist. In order to understand this effect we vary the minimum resolution of the simulation and calculate the velocity dispersion over time. The volume weighted velocity dispersion is defined as the standard deviation of the spatial velocity distribution. We calculate the total velocity dispersion by taking the square root of the sum of the variances of the spatial components. For the density weighted velocity dispersion we normalise the velocities with the density of its respective cell divided by the average density before calculating the variance. Note that we use the central axis of the box to define the centre of the filament. As we use a cartesian grid code this can lead to a wrong split-up in the cylindrical components if the filament axis lies not exactly in the centre of the box. We analysed the error in the total velocity dispersion using a cartesian and a cylindrical calculation and the difference is at most one per cent and therefore negligible. We show the evolution of the density weighted velocity dispersion in Figure 2. One can see that an equilibrium is established after about two Myr where the velocity dispersion becomes almost constant. We observe this behavior in all our simulations, with and without initial perturbation, and it is also found in turbulent smooth-particle-hydrodynamical simulations of filaments forming in a turbulent medium (Clarke et al., 2017). Varying the minimum resolution, one can see that the equilibrium level of the simulations including a perturbation decreases for higher resolution showing no more substantial change going from $512^{3}$ to $1024^{3}$ in minimal resolution. In contrast, the simulations without an initial perturbation develop higher and higher values of turbulence the better the resolution and begin to converge to a similar value as the cases with an initial perturbation. Moreover, even the visual impression of the turbulent structure for the cases with and without initial perturbations becomes increasingly similar with higher resolution. This is shown in Figure 3 where we compare the cases of highest minimal resolution. We interpret this in such a way that there is a inherent physical level of generated turbulence from accretion. As long as there is a source of perturbation the growth in turbulence will tend to this value. In the case of a smooth initial inflow the perturbation is given by numerical noise which is smeared out for lower minimal resolutions due to numerical viscosity. For higher resolution, numerical viscosity is low enough that the numerical noise can grow to the same level of turbulence as in the case containing an added perturbation. The interstellar medium is not completely smooth and will always contain density fluctuations. In order to validate if the generated velocity dispersion depends on the initial density perturbation we vary the initial density perturbation in strength and form for a constant minimum resolution. Using a flat, "white noise" perturbation on the one hand and a Gaussian perturbation on the other shows no quantitative difference in magnitude of the intrinsic filament velocity dispersion. Varying the perturbation strength also shows no dependence on the initial perturbation. In Figure 4 we show that even changing the perturbation amplitude over five orders of magnitude, the resulting value of velocity dispersion varies only minimally. As there is no more significant change for a minimum resolution of $1024^{3}$ all following analysis is carried out for a minimum resolution of $512^{3}$.

We also evaluate the radial evolution and mass accretion rate of the filament including an initial perturbation in order to verify if Equation 7 and Equation 10 hold. In Figure 5 we show both of them together with the analytical expectations. As our analytical prediction, the measured mass accretion rate is constant. Despite being close to the predicted value given by the blue dashed line, it has a small but measurable constant offset. This effect can be explained if we add the increase in mass accretion given by the expansion of the filament. Adding the radial growth velocity to Equation 7 gives the blue dotted line. This effect itself also should lead to a faster radial growth but the effect is minimal as we cannot see an significant offset of the analytical expectation for filament radius $R(t)$. We also consider that radial turbulent motions could increase the radial growth. Correcting the velocity for the radial expansion given by Equation 10 as $2c_{s}^{2}/v_{r}$ by including the contribution of the density weighted radial turbulence as $2(c_{s}^{2}+\sigma_{r}^{2})/v_{r}$ and adding this term to the inflow velocity in the mass accretion rate Equation 7, gives an overestimation for the radial growth and a mass accretion rate (dashed-dotted lines in Figure 5). At about 3.5 Myr the filament reaches the limits of the box and its maximum numerically resolved extent. This means that mass cannot be effectively accreted and the mass accretion rate goes to zero which leads to a decay in velocity dispersion seen in later evolution plots.

We show a detailed analysis in the next subsections where we present the cases of an inflow velocity generating a super- and subsonic internal velocity dispersion. This leads to the following conclusion: As long as one includes a symmetry breaking perturbation, either numerical or artificial the amount of generated turbulence is robust as long as the minimal resolution is high enough.

4.2 Trans- and supersonic turbulence

In Figure 6 we plot the evolution of a Mach 5.0 inflow with an initial density perturbation. The filament shows every indication of turbulent motions rearranging material constantly. The bubbling and sloshing in the filament forms temporary ridges and overdensities, the most prominent on the central line at the beginning of the simulation. Over time the central overdensity weakens as the lack of gravity allows the material to spread freely. Nevertheless, the visual impression is that of an observable filament albeit the filament broadens to an unrealistic width. The velocity dispersion settles to an equilibrium as shown in Figure 7. The volume weighted velocity dispersion is in the supersonic regime and dominated by the radial velocity dispersion. Interestingly, the longitudinal and angular velocity dispersions, both volume and density weighted, settle to the same level of about half the sound speed. Furthermore, their values are at about 2/3 of the radial and about half the total velocity dispersion. This relation between the radial velocity dispersion and the other components remains robust and true for all cases of generated velocity dispersion, not only in the supersonic case. As the filament reaches the boundary of the box earlier than the reference case, the velocity dispersion decays after 3.5 Myr. Additionally, we show the build-up of the kinetic energy power spectrum of the filament in Figure 8. As the filament grows, the maximum of the distribution shifts to bigger scales. It always follows closely the filament radius, given by the dashed vertical lines. Thus, the radius is the scale for the driving mechanism. Included in the plot are the predicted scaling relations for the power spectrum. In the supersonic regime, numerical studies (Kritsuk et al., 2007; Federrath et al., 2010; Federrath, 2013) have shown that a pure velocity power spectrum should follow Burgers turbulence (Burgers, 1948) with a scaling of $-2$. In the subsonic regime, Kolmogorov’s theory of incompressible turbulence (Kolmogorov, 1941) predicts a decay to smaller scales with a power law of $-5/3$. As one can see, the geometrical form of the filament limits the power on large scales. Nevertheless, the power spectrum follows the expectation quite well on intermediate scales. However, it is impossible to distinguish between supersonic and Kolmogorov decay as both of them lie too close together. In Figure 9 we show the volume weighted probability density function (PDF). As expected for fully developed, supersonic turbulence it follows a log-normal distribution (Vazquez-Semadeni, 1994; Padoan et al., 1997; Passot & Vázquez-Semadeni, 1998). The width of the distribution is correlated to the Mach number as

[TABLE]

The parameter $b$ depends on the ratio of solenoidal to compressional driving of the turbulent motions, varying smoothly from $b=1/3$ for purely solenoidal to $b=1$ for purely compressive driving with a value of $b\approx 0.4$ for a natural mix of modes $F_{\mathrm{comp}}/(F_{\mathrm{sol}}+F_{\mathrm{comp}})=1/3$ (Federrath et al., 2008, 2010). All of our simulations with supersonic turbulence show about the same value of $b\approx 0.55$ but there is also some variation over time.

4.3 Subsonic turbulence

In contrast to the turbulent motions of the Mach 5.0 inflow, a slower inflow velocity is only capable of generating subsonic turbulent motions despite itself being supersonic. In Figure 10 we show the visual impression of a Mach 3.0 inflow. It is not strong enough to generate substructure inside the filament. Only the surface is mildly perturbed. The lack of internal motions can also be seen in the velocity dispersion evolution in Figure 11. At all times it is below the sonic line. Nevertheless, it again reaches an equilibrium after about 2.0 Myr which decays after reaching the size of the box at about 3.4 Myr. We also show the kinetic energy power spectrum for the subsonic turbulent case in Figure 12. The power spectrum shows a similar behavior to the supersonic case and again the maximum of the cascade, the indicator of the driving scale, corresponds to the filament radius. As in the supersonic case, it is not easy to distinguish between a Kolmogorov and an $k^{-2}$ cascade but at later times the power law seems more similar to the former one which is to be expected for subsonic turbulence. In contrast to the supersonic case, we see a strong lack of compressional modes in the split-up of the power spectrum. This is also confirmed by Equation 18 where we get a value of $b=1/3$.

4.4 Dependence on inflow velocity

The crucial question of accretion driven turbulence is how much internal velocity dispersion in a filament is generated in dependence of the inflow velocity. Therefore, we repeat the simulation for inflow velocities of Mach number 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 5.0, 6.0, 7.5 and 10.0 for a minimum resolution of $512^{3}$. We show the resulting values of the velocity dispersion in Figure 13 where we plot them against their inflow velocity. The errorbars are given by the inherent variance of the velocity dispersion for different initial seeds of the perturbation spectrum and is about five per cent of the measured turbulence. In Figure 14 we also show the dependence of the Mach number squared on the inflow Mach number squared as this is the measure of the turbulent and inflowing energy. Both the accreted kinetic energy and the turbulent energy depend on the total accreted mass of the filament. Nevertheless, one does not have to account for mass as all the accreted material also has to be set into turbulent motions and thus the mass term cancels from both energy terms. This is illustrated later in Equation 24. In both figures we show the volume weighted velocity dispersion on the left hand side and the density weighted velocity dispersion on the right hand side. One can see that the measurement method has a major impact on the results. First, we look at the volume weighted velocity dispersion in Figure 13. We observe three distinct regimes: A high inflow velocity (light blue) leads to supersonic turbulence, an intermediate inflow velocity (medium blue) generates subsonic turbulence and a low inflow velocity (dark blue) results in nearly no turbulence. All of these regimes can be well fitted by linear relationships which do not necessarily go through the zero point. The fitting parameters are always given in the legend. Striking is the break at the sonic line (dashed horizontal line) going from the intermediate subsonic regime to the high velocity supersonic regime. This break does not appear in the density weighted velocity dispersion on the right hand side of Figure 13. Now the high and medium inflow velocity values follow one and the same relation. Thus, we now can fit the data by one single linear relationship that connects both regimes and that is shown by the dashed orange line with the parameters:

[TABLE]

The fact that we can fit a single line from the high supersonic end down to low subsonic turbulent velocities shows that there is no difference in the physics at work in both regimes. The break in the volume weighted relation also occurs for the squared volume weighted velocity in Figure 14. Although the slope stays roughly the same there is a gap between the supersonic and the subsonic regime. Contrarily, the density weighted velocity dispersion squared shows a smooth transition from nearly no turbulence to a constant slope. The density weighted velocity dispersion is also a measurement of the kinetic energy in the turbulence of the filament, a fact that we have confirmed by calculating the kinetic energy. The break in the volume weighted velocity dispersion results from the fact that the nature of the velocity and density distribution changes from the subsonic to the supersonic regime. Supersonic turbulence is shock dominated and due to compression most of the kinetic energy is in high densities. The information about the high densities is lost if one only measures the volume weighted velocity dispersion.

In the right panel of Figure 14 we also overplot Equation 19. Despite being a linear correlation between velocity dispersion and infall speed it fits the numerical results remarkably well and deviates only slightly for very low Mach numbers. We also attempt to fit a parabola to all of the density weighted data but are not able to get a good match. Therefore, we conclude that the linear relation of Equation 19 provides the best analytical description of the underlying physics. While the data points could also be fitted reasonable well with an offset power law of 1/3rd, they do not follow the simple prediction of Heitsch (2013), given by Equation 16. We discuss the implication of this in section 7.

We repeat a subset of the simulations for a higher temperature of 40 and 100 K and get similar results, all lying on the analytical relation given by Equation 19. Thus our results are independent of the temperature. We also repeat a subset of the simulations with an additional longitudinal velocity component in the inflow. The result in velocity dispersion is unchanged and the only difference is that the whole filament now begins to move in x-direction with constant velocity.

5 Pressure equilibrium

In order to see the impact of turbulence on the pressure we analyse its components as function of radius and time. In Figure 15 we show the different contributions to the pressure calculated in radial bins for the Mach 5.0 case with an initial density perturbation. We calculate the velocity dispersion, the average density and average thermal pressure in each slice along the filament and in radial bins which are chosen to be four cells wide. We use these values to calculate the respective pressure terms and take the mean along the filament in order to determine an average value for the whole filament. There is an overlap region where filament gas and environment gas mix due to the filament not being completely straight and round. Therefore, we only use the filament gas to calculate the pressure components in the overlap region. We determine the turbulent pressure component with the density weighted velocity dispersion. Thus, the turbulent pressure is given by

[TABLE]

where the braket notation represents the expectation value. The average ram pressure is calculated by the average density and radial velocity as:

[TABLE]

One can see in Figure 15 that outside of the filament the turbulent pressure (dashed-dotted line) is zero and the thermal pressure (dashed line) is negligible. As material streams towards the filament the ram-pressure (dotted line) increases as the density increases. When the accretion flow reaches the filament the ram-pressure breaks down and some of the energy is converted into turbulent motions giving rise to a turbulent pressure component. In the overlap region of filament and environment gas we see both a contribution of turbulent pressure and ram pressure due to our split-up of components. As one can see the average pressure inside the filament given by the average thermal pressure together with the average turbulent pressure (solid line) has nearly the same value as the average ram pressure. The filament is compressed and restrained by the accretion flow. Before the filament has settled there is quite an overpressure in the centre of the filament. As the filament grows outward, it adjusts to the outside ram-pressure and a constant pressure inside the filament is established. Note that there is a gradient in the pressure components inside the filament where the thermal pressure decreases and the turbulent pressure increases towards the edge of the filament. Most of the turbulent motions are created at the boundary and propagate inwards. This also explains why there is no turbulent component supporting the filament radially as the turbulence acts in a localized region.

6 Observable velocity dispersion

In order to compare our simulations to observations we derive the line-of-sight velocity by sending a ray through the computational domain. We vary the inclination of the filament and treat every crossed volume element with its respective density as a discrete emitter of a line-of-sight velocity value. These are converted to density weighted Gaussian line profiles with a dispersion width corresponding to the thermal linewidth of  C18O at 10 K with a given value of $\sigma=0.0526$  km s*-1*. We bin the resulting line profiles into histograms with a bin width of $0.05$  km s*-1* to get a complete line emission for each observed spatial pixel. We then measure the velocity centroids and the linewidths as an observer would do by fitting Gaussians to the line. As several line profiles show multiple components in velocity space, we fit multiple Gaussians to a single line. In contrast, fitting only one Gaussian to a line overestimates the velocity dispersion as multiple components are grouped together to form one broad line. To prevent false detections we discard peaks which are smaller than a quarter of the maximum line profile and closer together than 1.5 times their respective width. To obtain the non-thermal velocity dispersion we have to subtract the contribution of the thermal gas motions of the full width half maximum (FWHM) of the line as e.g. Myers (1983):

[TABLE]

We show the non-thermal linewidth along a filament through its centre for the case of turbulence forming due to inflows of Mach 3.0, 5.0 and 10.0 in Figure 16, Figure 17 and Figure 18, all of which include an initial perturbation. At a first glance it is important to note the striking similarity to real observations (Hacar & Tafalla, 2011; Hacar et al., 2013; Tafalla & Hacar, 2015; Hacar et al., 2016). As in the actual measurements we see an inherent spatial variation due to the non-homogeneous nature of the turbulence. The variation is also not stationary but changes with time, mimicking the sloshing motions of the total filament. However, the mean of the data distribution stays relatively stable, only varying by about 10%. Comparing the mean to the analytical value of the velocity dispersion it becomes clear that an observer cannot see the full picture of the internal motions of the filament. The observer will preferentially see the density weighted radial velocity dispersion (dashed line) as filaments are more likely to be detected for small inclinations. Thus, the line-of-sight is parallel to the radial motions. While the subsonic case shows mean values which lie below any velocity dispersion typically observed, it is also the only one where the mean of the observed distribution lies above the expected value of the radial velocity dispersion. Going to higher inflow speeds, the mean of the observed distribution generally lies below the expected value. The situation becomes even worse for cases of high turbulence as shown in Figure 18 where the discrepancy can be even as large as a factor of two. Moreover, the discrepancy between the observed mean and the total velocity dispersion is even larger. Despite having a supersonic total density weighted velocity dispersion, most of the data points of the Mach 10 inflow show a subsonic or at best a transsonic non-thermal line width. This means that the interpretation of the inherent motions in an observed filament can be severely flawed. A filament dominated by supersonic motions will be classified as only having subsonic motions. With the addition of gravity the problem will become even worse as there is an internal density gradient enhancing the signal of the central region. This leads to an overall thinner line as the outer region, which contains a high radial velocity component, is neglected and thus the measured line width becomes even smaller. There is not even a constant correction factor as for example assumed for isotropic turbulence of $\sqrt{3}$. The measured mean is relatively constant for different inclinations, going from zero to sixty degrees, but if one does not split the individual components in the line its broadness is overestimated the more the filament is seen from the side. The larger the inclination, the more is the filament dominated by longitudinal motions which give only one thin component. Therefore, the observable mean decreases and finally approaches the longitudinal velocity dispersion. This can be seen to some degree in the subsonic case. If one fits one component to the line-of-sight this effect is stronger and can also be observed in the cases of higher inflow velocity. It is even possible for the mean to be below the longitudinal velocity dispersion as a single line-of-sight does not contain the complete information on the total line-of-sight velocity dispersion of the whole filament. Striking is also the fact that the variance of the distribution decreases for a higher inclination. This behavior could potentially be used to determine the inclination of a filament. We will study the effect of the inclination angle on the measurements of the turbulent velocity and its variance in a subsequent paper.

7 Discussion and conclusions

We have presented a numerical study on accretion driven turbulence in filaments. The focus of this paper was to analyse the dependence of the velocity dispersion on the inflow velocity. We deliberately neglected the effects of gravity in order to have a constant inflow velocity and to be able to follow the driving of turbulence long enough without the radial collapse of the filament or its condensation into collapsing clumps. The formation of turbulent filament, including self-gravity will be discussed in a subsequent paper. We find that there is a linear dependency of the density weighted velocity dispersion on the inflow velocity. Below Mach 2.0 the relationship flattens as shown in the right panels of Figure 13 and Figure 14. Why there is a break in the relation is not completely clear. One possibility is that the low inflow velocity region is an artifact of numerical noise. In this case it must be dominated by perturbations on small scales. Therefore, we smooth our data to remove the small scale signal and recalculate the velocity dispersion. The result shows that the velocity dispersion is unchanged. Therefore, the measured velocity dispersion is in the large scale structure. Even for a high degree of smoothing the velocity dispersion remains the same. Thus, there must be a physical reason for the break in the slope. An explanation could be that filaments have a bottom level of minimal velocity dispersion comparable to a basic eigenmode which is exited by the constant inflow. In the subsonic regime sound waves are supposed to be very inefficient in dissipating energy and indeed if one sets up a filament with a random velocity perturbation there is a level of velocity dispersion below which the kinetic energy does not decay. Its dependence on form and density of the filament is out of the scope of this paper and will be discussed elsewhere.

We now want to focus on the form of the linear fit. As shown in Figure 13 we measure a linear relationship of

[TABLE]

It is possible to show that this is equivalent to an energy balance. In order to get a kinetic energy we take the square of the equation and multiply it with half of the mass accretion rate which gives

[TABLE]

with the restriction that

[TABLE]

For a constant velocity dispersion the term on the left hand side is the change in turbulent energy and the first term on the right hand side the rate of accreted kinetic energy. To show the meaning of the remaining terms we use the equation of change in turbulent energy given by

[TABLE]

where $\dot{E}_{t}$ is the change in turbulent energy, $\dot{E}_{a}$ is the kinetic energy accretion rate and $\dot{E}_{d}$ is the energy dissipation rate. In the case of a constant velocity dispersion, the change in turbulent energy is due to the change in total mass of the filament. The same is true for a constant accretion velocity where the change in total accreted kinetic energy is due to the change in the total accreted mass. We also consider that energy is lost in the isothermal accretion shock and is radiated away. As the fraction of the lost energy should be roughly constant per time we introduce a constant efficiency $\alpha$ which has a value between zero and one. Comparing Equation 26 to Equation 24 we determine the efficiency to be:

[TABLE]

This means that the energy lost to dissipation is given by the remaining terms

[TABLE]

A possible explanation for both terms is the assumption that turbulence is mainly dissipated in an inner region of the filament where radial waves travelling inwards interact. We assume that this region is proportional to the total radius of the filament:

[TABLE]

The proportionality $b$ has to be constant for a single run but can vary with accretion velocity. The mass contained in the inner region is then the density given by the shock times the volume:

[TABLE]

This means that the mass contained in a region proportional to the radius is also proportional to the total mass. Thus, if we use the definition of the energy dissipation rate given by Equation 14, assume that the scale relevant for the crossing time is the radius of the inner region and assume that the speed for the crossing time is the sound speed we get:

[TABLE]

As the inner region should be larger for greater inflow Mach number $\mathcal{M}$ we assume that:

[TABLE]

where $a$ is a constant. Thus the energy decays as:

[TABLE]

If we compare our result to Equation 28 we find this fits both terms with $a=0.72$.

Our results can be summarized as follows:

The accretion of material leads to non-isotropic, sub- and supersonic turbulence depending on the energy accretion rate as long as the symmetry is broken. The efficiency of transferred energy is independent of the accretion rate and equal to 9%. 2. 2.

The amount of turbulence generated is independent of the perturbation strength of the initial density field and independent of an additional velocity component parallel to the filament. 3. 3.

The turbulence reaches an equilibrium level which scales linearly with the accretion rate. 4. 4.

The filament radius grows linearly with time and is the driving scale of the turbulence. 5. 5.

The density weighted velocity dispersion contributes to the pressure equilibrium inside the filament. 6. 6.

In most situations, an observer cannot measure the true velocity dispersion and will even misinterpret the level of turbulence to be subsonic while the intrinsic velocity dispersion could still be supersonic.

Acknowledgements

We thank the whole CAST group for helpful comments and discussions. AB, MG and SH are supported by the priority programme 1573 "Physics of the Interstellar Medium" of the German Science Foundation and the Cluster of Excellence "Origin and Structure of the Universe". The simulations were run using resources of the Leibniz Rechenzentrum (LRZ, Munich; linux cluster CoolMUC2). We would also like to thank the referee whose input lead to an improvement of the results of the paper.