The Correlation Between Magnetic Pressure and Density in Compressible MHD Turbulence

A&A 398, 845–855 (2003) Astronomy DOI: 10.1051/0004-6361:20021665 & c ESO 2003 Astrophysics

The correlation between magnetic pressure and density in compressible MHD turbulence

T. Passot1 and E. V´azquez-Semadeni2

1 CNRS, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice Cedex 4, France e-mail: [email protected] 2 Instituto de Astronom´ıa, UNAM, Unidad Morelia, Apdo. Postal 7-32, Morelia, Michoacán 58089, Mexico e-mail: [email protected] Received 7 August 2002 / Accepted 15 October 2002 Abstract. We study, both analytically and numerically, the behavior of magnetic pressure and density fluctuations in turbulent isothermal magnetohydrodynamic (MHD) flows in a slab geometry. We first consider “simple” MHD waves, which are the nonlinear analogue of the slow, fast and Alfvén linear waves, and show that the dependence of magnetic field strength B on 2 density ρ in a simple wave depends on the mode which is considered: for the slow mode, B c1 c2ρ, while for the fast mode, B2 ρ2. We also perform a perturbative analysis about a circularly-polarized plane Alfv´'en wave− to investigate Alfvén ' wave pressure, recovering the results of McKee and Zweibel that B2 ργe , with γ 2atlargeM , γ 3/2 at moderate M ' e ' a e ' a and long wavelengths, and γe 1/2atlowMa. This wide variety of behaviors implies that a single polytropic description of magnetic pressure is not possible' in general, but instead depends on which mode dominates the density fluctuation production. This in turn depends on the angle θ between the magnetic field and the direction of wave propagation and on the Alfvénic Mach number Ma. Typically, at small Ma, the slow mode dominates, and B is anticorrelated with ρ.AtlargeMa, both modes contribute to density fluctuation production, and the magnetic pressure decorrelates from density, exhibiting a large scatter, which however decreases towards higher densities. In this case, magnetic “pressure” does not act as a restoring force, but rather as a random forcing. These results have implications for the probability density function (PDF) of mass density. The non- systematic behavior of the magnetic pressure causes the PDF to maintain the log-normal shape corresponding to non-magnetic isothermal turbulence, except in cases where the slow mode dominates, in which the PDF develops an excess at low densities because the magnetic “random forcing” becomes density-dependent. Our results are consistent with the low values and apparent lack of correlation between the magnetic field strength and density in surveys of the lower-density molecular gas, and also with the recorrelation apparently seen at higher densities, if the Alfvénic Mach number is relatively large there. Key words. magnetohydrodynamics – turbulence

1. Introduction been investigated in a variety of scenarios. Since observations often suggest that there is near equipartition between turbu- The cold molecular gas in our Galaxy is generally believed to lent and magnetic energies in molecular clouds (e.g., Myers be magnetized and turbulent (see, e.g., the reviews by Dickman & Goodman 1988; Crutcher 1999), the motions are often con- 1985; Scalo 1987; Heiles et al. 1993; Vázquez-Semadeni et al. sidered to be MHD waves (Arons & Max 1975). Until re- 2000). However, at present, the actual strength of the magnetic cently, Alfvén waves were favored, as these were expected to field in molecular clouds and, as a consequence, its dynamical be less dissipative than their slow and fast counterparts, but re- importance, are still a matter of debate, with opinions ranging cent numerical simulations (Mac Low et al. 1998; Stone et al. from considering it crucial (e.g., Crutcher 1999) to moderate 1998; Padoan & Nordlund 1999) have shown that supersonic (e.g., Padoan & Nordlund 1999), although in general, the ob- MHD turbulence decays almost as rapidly as purely hydrody- servational evidence so far appears inconclusive (e.g., Bourke namic turbulence, both incompressible and compressible (Cho et al. 2001; Crutcher et al. 2002). Therefore, it is important to & Lazarian 2002). An exception to fast decay is the case of continue gathering information both observationally and theo- unbalanced turbulence discussed, e.g., by Cho et al. (2002a,b). retically on the distribution and strength of the magnetic field The rationale for the preponderance of the Alfvén mode be- in the ISM and, in particular, molecular clouds. ing less clear now, it appears necessary to investigate all MHD In general, magnetic fields and turbulence are believed modes as a source of pressure in a general context in fully tur- to provide support against the self-gravity of the clouds, bulent regimes. which typically have masses much larger than their thermal Jeans mass. In this context, the pressure provided by The functional form that the magnetic pressure may take magneto-hydrodynamic (MHD) motions in the clouds has as a function of density is also important for the statistics Send offprint requests to:E.Vázquez-Semadeni, of the density fluctuations. It was indeed shown by Passot & e-mail: [email protected] Vázquez-Semadeni (1998; see also Nordlund & Padoan 1999)

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20021665 846 T. Passot and E. Vázquez-Semadeni: Density-magnetic pressure correlation that for flows in which the pressure P exhibits an effective poly- of thermal condensation triggered by strong compressions in γ tropic behavior, with P scaling with density ρ as P ρ e , γe the presence of a uniform magnetic field, which are in several being the effective polytropic exponent, the shape of∼ the den- aspects similar to gravitational collapse, Hennebelle & Pérault sity probability density function (PDF) depends on the specific (2000) found that the magnetic field does not necessarily in- value of γe. The PDF takes a log-normal form for γe = 1, but it crease together with the density. develops an asymptotic power-law tail at low (high) densities Recent theoretical work (e.g., Lithwick & Goldreich 2001; for γe > 1(γe < 1). Thus, the value of γe produced by the var- Cho et al. 2002) has not specifically addressed the issue of the ious sources of pressure is important for the density statistics magnetic pressure-density correlation, as it has focused mainly and the overall dynamics of the flow. on the spectral properties of moderately compressible MHD In an analysis of the pressure produced by Alfvén waves, turbulence, as a consequence of mode coupling. Nevertheless, McKee & Zweibel (1995, hereafter MZ95) concluded, based Cho et al. (2002) mention in passing that their results lead them on an analysis by Dewar (1970), that it is isotropic, and that, to not expect a significant correlation between the magnetic for flows undergoing slow compressions, γe = 3/2, while for pressure and the density. strong shocks, γe = 2. Vázquez-Semadeni et al. (1998) stud- In an attempt to understand the physics underlying the ied the evolution of the velocity dispersion σ in gravitation- above experimental and observational results, in this paper we ally collapsing flows, as a means of determining the density study the dependence of magnetic pressure on density as a con- dependence of the “turbulent pressure” Pturb, defined in such a sequence of nonlinear MHD wave propagation in isothermal way that σ2 dP /dρ. They found that for slowly collapsing MHD turbulence in “1+2/3” dimensions, also referred to as a ≡ turb magnetized cases, γ 3/2, while for rapid collapse, γ 2. “slab” geometry. This choice, where all three components of e ∼ e ∼ More recent numerical work has directly plotted the the velocity and magnetic fields are integrated along one sin- magnetic strength B or the magnetic pressure B2 in gle space dimension, allows us to isolate the effects of the rel- three-dimensional numerical simulations of both isothermal ative orientation of the magnetic field and wave propagation (Gammie & Ostriker 1996; Padoan & Nordlund 1999; Ostriker directions. We first consider the problem analytically, finding et al. 2001) and non-isothermal flows (Kim et al. 2001; asymptotic relations between the density, magnetic field and Mac Low et al. 2001). The isothermal simulations should velocity field fluctuations, which show the effectiveness of the be reasonable models of molecular clouds, while the non- slow and fast modes as sources of density fluctuations under a isothermal ones are cast as models of the ISM at large. Gammie variety of conditions (Sect. 2). We then discuss Alfvén wave & Ostriker (1996) report an almost constant (on average) pressure using a perturbation analysis, comparing with the re- magnetic field strength as a function of density in non-self- sults of MZ95 (Sect. 3). These results are then tested by means gravitating runs. Padoan & Nordlund’s (1999) plot of B vs. ρ of numerical simulations in the same geometry (Sect. 4), which shows a scatter of over two dex in the low density range for support the scenario derived from the analysis of the equa- large values of the Alfvénic Mach number, with the scatter tions. Caution must be taken when generalizing these results decreasing towards larger densities. On the other hand, there to three dimensions where all wave propagation directions are is less scatter at low Alfvénic Mach numbers (Ma), but it is present. After a summary, this issue is discussed in Sect. 5.1. more uniform throughout the density range. Those authors note Implications for the interpretation of observational results are that the upper envelope of their B distributions closely matches evoked in Sect. 5.2. the observational scaling B ρ1/2 found in cases when the magnetic field is detected (e.g.,∼ Crutcher 1999; Crutcher et al. 2002), although they remark that in a large number of cases 2. Properties of simple waves only upper limits or complete non-detections are obtained. The equations governing, in the magnetohydrodynamic(MHD) Such scaling also corresponds to the high-B tail of the mag- limit, the one-dimensional motions of a plasma permeated by a netic strength distribution in the high-Ma simulation by Padoan uniform magnetic field B0 in a slab geometry read & Nordlund. It is worthwhile to additionally note that the bulk ∂ρ ∂(ρu) of the points in their B distribution does not follow a simple + = 0(1) power-law scaling with ρ, but instead appears to curve up, be- ∂t ∂x ing first nearly constant with ρ and then starting to rise. Ostriker ∂u ∂u 1 ∂ ρ b 2 + u = + | | + fx (2) et al. (2001) find similar results, except that these authors only ∂t ∂x −ρ ∂x M2 2M2 s a ! attempt to fit the high-density part of the B distribution to ob- ∂v ∂v b ∂b + u = x + f (3) servations, finding slopes between 0.3 and 0.5, which again are ∂t ∂x M2ρ ∂x not too different from the observed scaling. a ∂b ∂ ∂v In non-isothermal simulations, Mac Low et al. (2001) have + (ub) = bx , (4) plotted the magnetic pressure vs. the thermal pressure, finding ∂t ∂x ∂x again a large scatter of up to 6 orders of magnitude in their where x is the direction of propagation, nondimensionalized by supernova-driven simulations, while Kim et al. (2001) show a typical length L. The velocity components u (along the x axis) plots of B vs. ρ, finding slopes of 0.4, although with scatter of and v = vy + ivz (along the y and z axes) are normalized by a ∼ one order of magnitude at high density, and two orders of mag- velocity unit U0, the mass density ρ by a reference density ρ0, nitude at lower densities, the plot actually appearing more like and the magnetic field components bx = cos θ and b = by + ibz two segments with very different slopes. Finally, in simulations by B0. Note that bx is constant and that bz(t = 0) = sin θ,where T. Passot and E. Vázquez-Semadeni: Density-magnetic pressure correlation 847 b θ is the angle between the direction of propagation and that Vρ v x b d 2 d = 0(9) of the initially unperturbed ambient magnetic field. Time t is − − Ma measured in units of L/U0. An isothermal equation of state is Vdb + bdu bxdv = 0. (10) assumed and we denote by c the constant sound speed. Two − − s The system (7)–(10) has non-trivial solutions if the wave non-dimensional numbers can be defined, the sonic Mach num- speed V is given by ber Ms = U0/cs and the Alfvénic Mach number Ma = U0/va, 1/2 where va = B0/(4πρ0) is the Alfvén speed of the unperturbed 2 2 2 1 2 4βbxρ Ma V = B + βρ 1 1 (11) system. The plasma beta is here defined by β = 2 . The above 2 2 2 Ms ± 2Ma ρ  ± s − (B + βρ)  system of equations also contains a driving in the form of ran-     f u  bx  dom accelerations x (acting on the component of the veloc- or if it equals the Alfv´en speed VA = 1/2. Recall that we ± Maρ ity) and f = fy + ifz, (acting on the components v). 2 2 2 denote by B = bx + b the total magnetic intensity. In this section we shall derive some properties of the so- The latter root is| associated| with the circularly polarized called simple MHD waves (see, e.g., Landau & Lifshitz 1987, Alfvén simple wave, which is non-compressive and has b 2 = Sect. 101) and thus we assume fx = f = 0. In the case where const. The solutions (11) are associated with fast and slow| sim-| the basic state is perturbed by infinitesimal disturbances, the ple waves. The speeds of the linear fast and slow magnetosonic solutions are superpositions of linear plane travelling waves. waves are recovered by taking ρ = 1andb = i sin θ. These plane waves are monochromatic and their profile does We are thus led to the following system of equations for the not change in time, with all quantities only depending on the wave profiles combination x Ct. The propagation velocity C is a constant ± du V that identifies with one of the three possible roots of the dis- = (12) persion relation, namely the slow, Alfvén or fast velocity. For dρ ρ a particular plane wave solution each perturbed quantity can d b 2 d B2 | | = = M2V2 β . (13) be expressed as a function of a chosen one, for example the dρ 2 dρ 2 a − density ρ. Equation (13) can also be rewritten as dP/dρ = V2,whereP de- In the case of finite-size perturbations these relations do not b 2 ρ notes the total pressure | | 2 + 2 . Equations (11)–(13) can be hold, but it is nevertheless possible to look for particular solu- 2Ma Ms tions that have the property that all quantities are only functions solved numerically but it is advantageous to search for asymp- of any single one of them, as in the linear theory. These partic- totic solutions in some limits. First of all, when β = 0, i.e. in the ular solutions, which generalize the linear plane wave solutions absence of thermal pressure, it is found that V = 0 (the slow 2 B2 − to the case of finite disturbances, are called simple waves. wave does not propagate) and V+ = 2 , which is the Alfvén Ma ρ Following Landau & Lifshitz (1987, Sect. 105; see also speed based on the total magnetic field intensity. When bx = 0, Mann 1995), we recall here how to derive the relevant equa- i.e. for a propagation perpendicular to the ambient magnetic tions that determine simple wave profiles. The principle is il- 2 B2+βρ field, we again have V = 0 and now V+ = 2 . In both limits, − Ma ρ lustrated on the equation for mass conservation, that can be for the slow waves P = const., while for the fast waves B2 ρ2. rewritten in the form These relations are in fact more general as will be seen below.∝ ∂(ρ, x) ∂(ρu, t) In the case where 4βb2ρ (B2 + βρ)2 one obtains = 0, (5) x ∂(t, x) ∂(t, x) − b2 V2 = x (14) after transforming the partial derivatives into Jacobians. If one 2 2 − Ms (B + βρ) now chooses (t,ρ) as new independent variables, one has to B2 1 multiply the above equation by ∂(t,x) . Assuming that all depen- V2 ∂(t,ρ) + = 2 + 2 (15) dx Ma ρ Ms · dent variables only depend on ρ, and writing U = dt one gets, βρ after expanding the Jacobians, The above assumption only fails when 2 is of order unity to- bx b 2 du gether with | 2| small, i.e., when bx is not too small, for βρ U + ρ + u = 0, (6) bx − dρ of order unity and small field distortions. As an example, the above approximations fail for sound waves propagating along or, more generally, the magnetic field, for which arbitrary density pertubations can Vdρ + ρdu = 0, (7) develop on an unperturbed magnetic field. − Equation (13) together with the approximation (15) leads where V = U u denotes the wave speed. Each field is thus 2 2 − to B ρ for the fast wave. Using Eq. (14) for the slow speed, a function F(x (u + V)t)whereu and V are functions of e.g. one is∝ led to the following ordinary differential equation ρ. Each point of− the wave is traveling with its own velocity, 1 d b 2 b2 leaving the possibility for wave steepening and the subsequent | | = x 1, (16) formation of a discontinuity. 2β dρ b2 + b 2 + βρ − x | | Using the same procedure, Eqs. (2)–(4) read whose implicit solution reads 1 1 b 2 2 Vρdu + dρ + d| | = 0(8)b 2 2 2 2 2 βρ + | | + bx ln bx b βρ = C0, (17) − Ms Ma 2 2 | −| | − | 848 T. Passot and E. Vázquez-Semadeni: Density-magnetic pressure correlation

C0 being an arbitrary constant. For small values of bx,i.e.for (i.e. small values of ∆U ). Thus, at low values of β, density quasi-perpendicular propagation, the solution has a region at fluctuations are mostly| created−| by the slow mode, whose speed 2 small density where b C00 2βρ (constant total pressure), is then close to that of the sound wave. Conversely, at large β, | | ≈ − 2 2 followed by an asymptotic range where b = bx βρ.This the fast mode, which is essentially a sound wave, is the most | | − 2 behavior is still present when bx is closer to unity if b is effective one at generating density fluctuations. For interme- large enough and it is also obtained from Eq. (13) by numerical| | diate values of β, conclusions may depend on the magnitude integration. of bz. For example, for parallel propagation and large field dis- In order to interpret the numerical simulations of Sect. 4, it tortions, the slow mode tends to be the most efficient to produce is also useful to investigate under which conditions do the slow density fluctuations, except at large density. or the fast waves dominate the production of density fluctuations. A criterion that can be used is the relation between the total velocity vector U = (u,vy,vz) of the perturbation (in some 3. Alfven´ wave pressure way related to the fluid displacement) and the density perturbation. From Eqs. (7)–(11), it follows that, for small enough In the previous Section we considered waves propagating on perturbations, the density fluctuation ∆ρ and the velocity fluc- a uniform background. In a turbulent medium, the situation is usually more complex, as all types of waves are mixed. tuation ∆U = (∆u , ∆vy , ∆vz ) are related by ± ± ± ± Another case that still remains simple enough to be analyt- ∆ρ ically tractable, consists in studying the properties of MHD ∆u = V , ± ± ρ waves propagating on top of a circularly polarized, parallel- propagating Alfvén wave. These Alfvén waves are exact solu- and tions of the MHD equations (Ferraro 1955) and can be taken b b ∆ρ ∆v = x (V2 M2 β) of arbitrary amplitude. They read b0 = B0 exp [ iσ(k0 x ω0t)] 2 2 a − − ± −V Ma b ± − ρ · with ρ0 = 1, bx0 = 1, u0 = 0andv0 = b0/Ma. The polariza- ± | | tion of the wave is determined by the parameter− σ, with σ =+1 These relations are not valid for circularly polarized Alfvén (σ = 1) for a right-handed (left-handed) wave. The dispersion waves where b is constant. Using Eq. (11), it is easy to ver- − | | relation reads ω0 = k0/Ma. ify that ∆U+ ∆U = 0. In the linear case, i.e. when by = 0, · − Let us now consider perturbations of the form bz = sin θ and ρ = 1, one finds that, in the limit β 0 → iσ[(k+k )x (ω+ω )t] ∆U sin θxˆ cos θzˆ B0 b = b + b e− 0 − 0 + ∝ − ⊥ 0 + ∆U cos θxˆ + sin θzˆ B0 +iσ[(k k )x (ω ω )t] ( − ∝ k +b e − 0 − − 0 (23) while in the limit β − iσ(kx ωt) →∞ ρ = 1 + ρ0e− − + c.c. (24) ∆U+ xˆ k ∝ k iσ(kx ωt) ∆U zˆ k u = u0e− − + c.c. (25) ( − ∝ ⊥ iσ[(k+k0)x (ω+ω0)t] where k is the wavevector of the linear wave. v = v0 + v+e− − 2 2 Denoting X = V Ma and using the dispersion relation ± +iσ[(k k0)x (ω ω0)t] 2 2 ± 2 +v e − − − . (26) ρX (B + βρ)X + βbx = 0, one finds ± − ± − 2 2 1 2 2 ∆ρ The linearized equations read ∆U = B X + βρX 2βbx (18) | ±| M2 b 2 ± ∓ − ρ · a ! | | ωρ0 ku0 = 0 (27) 2 2 B βbx − In the limit β 0 one has X+ = and X = 2 so that k k ρ − B → ωu0 = ρ0 + (b0b∗ + b0∗b+) (28) M2 2M2 − 4 2 s a 2 B ∆ρ ∆U+ = (19) k0b0 ω0b0 1 2 2 (ω + ω0)v+ + u0 ρ0 = − (k + k0)b+ (29) | | ρMa b ρ 2 ! Ma − Ma Ma | | 2 β ∆ρ k b ω b 1 2 0 0 0 0 0 0 ∆U = 2 (20) (ω ω0)v u ∗ + ρ ∗ = − (k k0)b (30) | −| M ρ − M M 2 − a ! − − a a Ma − 2 (ω + ω )b (k + k )b u = (k + k )v (31) bx 0 + 0 0 0 0 + while in the limit β , X+ = β, X = and − − ρ 0 →∞ − (ω ω0)b (k k0)b0u ∗ = (k k0)v . (32) − − 2 − − − − − 2 β ∆ρ ∆U+ = (21) After some algebra, we obtain the following dispersion relation | | M2 ρ a ! 2 β2ρ ∆ρ 2 k2 k2 k2B2 ∆U 2 = (22) k 2 + 2 2 0 2 2 2 ω 2 ω+ −2 ω + 2 | −| Ma b ρ · M − M − M − − M | | ! s ! a ! a ! a 2 2 Relation (20) shows that when β is small, large density fluc- k 3 k 2 k0 kk0 ∆ρ ω ω + ω 3 ω + = 0. (33) tuations ( 1) can be created by small fluid displacements × − M M − M2 M3 ρ ∼ a !  a a a      T. Passot and E. Vázquez-Semadeni: Density-magnetic pressure correlation 849

Three different limiting cases can be considered and easily 4. Numerical simulations identified after rewriting the dispersion relation as In order to test the range of validity of the analytical predic- 2 2 k tions, we have performed numerical simulations of Eqs. (1)–(4) ω = 2 Ms in a periodic domain using a pseudo-spectral method based 2 2 on Fourier expansions. In order to numerically handle the k 3 k 2 k0 kk0 2 2 ω M ω + M ω 3 2 ω + 3 k B0 a a Ma Ma formation of strong shocks, dissipative terms of the usual − − 2 2 + (34) µl ∂ u µt ∂ v 2 2 2 form, namely and , are added to the right-hand-side Ma k+ 2 k 2 · ρ ∂x2 ρ ∂x2 2 ω+ −2 ω 2 Ma − Ma − − of Eqs. (2) and (3), respectively, and a magnetic diffusion η ∂ b ∂x2 In the case M , the dispersion relation approximates to the right-hand-side of Eq. (4). In addition, it was found nec- a 2 2 k2 B→∞2 ∂ ρ 2 k 0 essary to add a mass diffusion in the form µr 2 to the right- to ω 2 + 2 . If one assumes a polytropic dependence of ∂x ≈ Ms Ma hand-side of Eq. (1). This term preserves mass conservation the magnetic pressure on the density in the form b 2/2 ργ and and allows handling of strong shocks. If kept small enough, linearizes Eqs. (1)–(2), a direct comparison of the| | resulting∝ dis- it does not significantly modify the dynamics and in particu- persion relation with the above approximation leads to γ = 2. lar it does not alter the statistical conclusions we shall present. This is the case, mentioned by MZ95, in which the perturbation Except for one set of simulations discussed below, the forcing is very rapid compared to the speed of the Alfvén wave. This (actually an acceleration) acting on these equations is applied behavior of the magnetic pressure is easily analyzed directly on Fourier modes 1–19, peaked at wavenumber 8 with ampli- from the orginal MHD equations. In this case, in which the tudes proportional to k4 exp( k2/32) and Gaussian-distributed background magnetic field is weak, magnetic tension is negli- random phases. A forcing is− also applied on mode 0 in or- gible compared to field stretching. The magnetic field pertur- der to ensure momentum conservation. The random phases are bations are rapidly amplified and a balance is reached where 1 changed every 0.003 time units. A state of constant density and v = O( ), b = O(Ma), u = O(1) for temporal and spatial Ma zero velocity and magnetic field fluctuations is taken as the ini- scales both of order unity as well. In this situation, the term on tial condition. A resolution of 4096 grid points is used and typi- the right-hand-side of Eq. (4) can be dropped and this equation cal values for the diffusion coefficients are µl = µt = η = 0.003 leads to 4 and µr = 6. 10− . In the simulations, three main parame- 2 × ∂ b ∂ ∂ ters are varied, namely the angle θ and the Mach numbers Ms | | + (u b 2) + b 2 u = 0. (35) ∂t ∂x | | | | ∂x and Ma, but only θ and Ma are actually important as we only consider the high-Ms limit. For each simulation, we compute Together with Eq. (1), one obtains / 2 1 2 rms values of the sonic Mach number M˜ s = Ms U¯ ,where ∂ ∂ the brackets (resp. bars) denote time (resp. spatial) averaging. + u ln b 2 2lnρ = 0, (36) D E ∂t ∂x | | − Similarly, the rms Alfvénic Mach number and the effective beta 1/2 ! 2 2 ˜ ρU ˜ Ma ρ oftheflowaredefinedasMa = Ma 2 and β = 2 2 . confirming the result γ = 2 in this case. B Ms B In the case Ma 0, i.e. in the case of a strong background Finally, the density fluctuations and field distortions are de- → 1/2 1/2 magnetic field, magnetic tension is dominant and the Alfvén δρ˜ 2 δ˜B 2 2 2 fined as ρ = (ρ 1) and B = (by + (bz sin θ) ) wave is very rapid. When ω 0andk k0, i.e. when the per- respectively. − − turbation is very slow, or “quasi-static”,≈ the dispersion relation D E D E 2 2 2 The simplest case corresponds to perpendicular propaga- 2 k k B0 1 approximates to ω 2 + 2 ,givingγe = 2 , also derived tion (θ = π/2) where only fast waves can propagate. In that ≈ Ms 4Ma by MZ95 using a WKB approximation. case, we expect magnetic pressure to be very well correlated 2 2 Finally, for Ma finite but k 0andω ω0,i.e.for with density, with a dependance of the form b ρ ,what- ≈ | | ∝ quasi-uniform perturbations, the dispersion relation approxi- ever the value of Ma. This prediction is confirmed by the sim- 2 3k2 B2 2 k 0 3 ulations, as seen in Fig. 1, which shows the logarithm of the mates to ω 2 + 2 ,givingγe = 2 and recovering the ≈ Ms 4Ma 2 2 prediction of MZ95 for this case. two-dimensional histogram h ln(ρ), ln( b /2Ma ) of the mag- | | ˜ Whereas the predictions based on the WKB approach are netic pressure versus density forh a run with Ma =i5.16, adding recovered in this purely one-dimensional linear analysis, it points from temporal snapshots taken every 0.1 time units from should be noted that they are probably not valid in a fully t = 100 to t = 104. More precisely we display in grey scale three-dimensional situation. For example, MZ95 argued that (with black (resp. white) denoting the smallest (resp. largest) the Alfvén wave pressure is isotropic. A linear stability analysis value) the logarithm of the number of points (in the space-time analogous to the previous one but performed in the long-wave sample) in a given interval of b 2/2M2 and ρ, plotting only the | | a limit and for perturbations propagating exactly perpendicular points where the histogram is larger than 90% of the maximum to the background Alfvén wave shows that the Alfvén wave ex- at fixed ρ. We also calculate an average dispersion histogram in erts a pressure different from the 1D case, thus invalidating the the following way. For each density ρ, we define the mean of isotropy result. This pressure can even be negative and one re- the magnetic pressure logarithm as covers the polytropic index γe = 3/2 only in the limit of large amplitude background wave and for β = 0 (Passot & Gazol, ∞ in preparation). p¯m(ρ) = xh ln(ρ), x dx/H (37) Z0 850 T. Passot and E. Vázquez-Semadeni: Density-magnetic pressure correlation

Fig. 2. Log-log scatter plot of the magnetic pressure versus density for two simulations with cos θ = 0.1. The parameters of the left panel a), corresponding to a case with forcing on all three velocity components ˜ ˜ ˜ δρ˜ δ˜B are Ms = 6.63, Ma = 0.478, β = 0.00685, ρ = 2.69, B = 0.323 andσ ˜ = 0.18. For the right panel b), where the forcing is applied on the v velocity components only, M˜ s = 7.27, M˜ a = 0.527, β˜ = 0.00728, δρ˜ δ˜B ρ = 3.55, B = 0.324 andσ ˜ = 0.059.

Fig. 1. Log-log scatter plot of the magnetic pressure versus density ˜ ˜ ˜ for a simulation with θ = π/2, Ms = 4.12, Ma = 5.16, β = 1.52, to be log-normal (Passot & Vázquez-Semadeni 1998). The rise δρ˜ = 0.623, δ˜B = 0.618 andσ ˜ = 0.054. The line segment has a slope ρ B of the left tail seen in Fig. 3b is thus expected in the present equal to 2. case. When the angle θ is still slightly smaller than π/2but and the dispersion as with Ma 1, the magnetic field lines undergo large fluctuations and both slow and fast waves contribute to the production 1/2 ∞ 2 of density fluctuations. As seen in Figs. 4 and 5a, magnetic σ¯ (ρ) = x p¯m(ρ) h ln(ρ), x dx/H (38) pressure and density are now positively correlated (magnetic 0 − Z ! pressure roughly follows a polytropic law with γ = 2, corre- ∞ sponding to the fast mode), but the dispersion is larger than in where H = 0 h(ρ, x)dx. An average dispersionσ ˜ is evaluated by taking the mean ofσ ¯ (ρ)forln(ρ) varying by 10% about its the case of a small Alfvénic Mach number, indicative of the ad- R central value. ditional contribution of the slow mode. As a result, the stirring As the angle θ is decreased from π/2, the behavior is very due to magnetic pressure is statistically non-correlated with the sensitive to Ma.WhenMa 1 the field lines are still almost density and the density PDF is expected to be much closer to a unperturbed but the density fluctuations are then mostly gen- log-normal, as seen in Fig. 5. erated by slow waves and thus very different from the case When the angle θ is intermediate between parallel and al- θ = π/2, for which they are due to fast waves. In that case most perpendicular propagation, the distinction between the again we expect the magnetic pressure to exhibit little scatter, small and the large Alfvénic Mach number cases is not as clear. as the field is only slightly perturbed, and to be anti-correlated As shown in Figs. 6a, b for runs with an angle θ π/4and ≈ with density, as the total pressure remains roughly constant. for M˜ a = 0.521 and M˜ a = 2.48 respectively, there is a larger This is indeed the case, as seen in Fig. 2a. where the distribu- scatter of the points for small to intermediate values of the den- 2 2 tion of b /2Ma and ρ has a small scatter. In addition, a clear sity at large Alfvénic Mach number. In this case a positive cor- anticorrelation| | is observed at high density when no forcing is relation between magnetic pressure and density is noticeable at applied on the x-component of the velocity (Fig. 2b). In the high density (Fig. 6b). case of purely perpendicular propagation the density structures For parallel propagation with perpendicular forcing, the be- are those typically observed in neutral turbulence with a poly- havior is again strongly dependent on the Alfvénic Mach num- tropic index γ = 2, i.e. rather flat-topped structures (plateaux) ber. When Ma 1, the magnetic field is strongly distorted. separated by shocks that keep interacting, with a pre-eminence The forcing excites slow modes which develop into shocks, in- of low density regions (Passot & Vázquez-Semadeni 1998). In side which large density clumps form. These clumps, located contrast, for nearly perpendicular propagation and Ma 1, at local minima of the magnetic field intensity, are separated by the density structures are composed of large peaks that oscil- regions of large magnetic pressure (Fig. 8a). As a result, they late while avoiding collision (Fig. 3a). The probability density can barely approach each other but rather oscillate about their function (PDF) of the density shown in Fig. 3b shows an ex- mean position, at a frequency close to that of the fast wave of tended tail at large density with a significant excess at small the local total magnetic field. The signature of the slow wave density. Since the field strength is roughly constant and inde- dominance is the rather small scatter and the anti-correlation pendent of ρ, the magnetic pressure term in Eq. (2) acts like between magnetic pressure and density in compressed regions, a random acceleration whose strength increases as ρ decreases as seen in Fig. 7a corresponding to a case with M˜ a = 1.76. In (see Fig. 2a). Whereas the density field of an isothermal gas a way similar to the case of small Ma and almost perpendicu- stirred by a random acceleration has a log-normal distribution, lar propagation, the density PDF shown in Fig. 8b, displays an when the acceleration depends on the density, the PDF ceases excess at small density. T. Passot and E. Vázquez-Semadeni: Density-magnetic pressure correlation 851

2 2 Fig. 3. Top a) snapshot of log10 ρ (solid) and log10 b (dotted) for the Fig. 5. Top a) snapshot of log10 ρ (solid) and log10 b (dotted) for the run of Fig. 2a. Bottom b) density PDF for the run of| | Fig. 2a in log-log run of Fig. 4. Bottom b) density PDF for the run of| | Fig. 4 in log-log coordinates. The dashed line shows a ﬁt of the curve with a parabola coordinates. The dashed shows a log-normal ﬁt. corresponding, in these coordinates, to a log-normal distribution.

Fig. 6. Log-log scatter plot of the magnetic pressure versus density for two simulations with forcing on the three velocity components and cos θ = 0.7. The parameters of the left panel a) are M˜ s = 2.90, ˜ ˜ δρ˜ δ˜B Ma = 0.521, β = 0.00375, ρ = 1.20, B = 0.473 andσ ˜ = 0.32. ˜ ˜ ˜ δρ˜ For the right panel b) Ms = 6.59, Ma = 2.48, β = 0.141, ρ = 1.43, δ˜B B = 9.62 andσ ˜ = 0.49.

At small values of Ma, the magnetic field undergoes mild Fig. 4. Log-log scatter plot of the magnetic pressure versus density for variations. The perpendicular forcing preferentially excites fast a simulation with forcing on the three velocity components, cos θ = waves but the slow waves that form by nonlinear interac- ˜ ˜ ˜ δρ˜ δ˜B tions are the most effective at producing density fluctuations 0.1, Ms = 5.73, Ma = 7.29, β = 1.53, ρ = 1.32, B = 2.17 andσ ˜ = 0.61. The line segment has a slope equal to 2. since, the field being almost straight, only thermal pressure acts against compression at dominant order (see below for the effect 852 T. Passot and E. Vázquez-Semadeni: Density-magnetic pressure correlation

Fig. 7. Log-log scatter plot of the magnetic pressure versus density for two simulations with θ = 0. The parameters of the left panel a), where the forcing only aﬀects the perpendicular velocity components, ˜ ˜ ˜ δρ˜ δ˜B are Ms = 2.72, Ma = 1.76, β = 0.440, ρ = 1.04, B = 4.73 andσ ˜ = 0.21. For the right panel b), where the forcing is applied on the three ˜ ˜ ˜ δρ˜ velocity components, Ms = 5.10, Ma = 2.36, β = 0.218, ρ = 1.73, δ˜B B = 7.91 andσ ˜ = 0.57.

Fig. 9. Log-log scatter plot of the magnetic pressure versus density for a simulation with θ = 0 and forcing on the perpendicular velocity components. The parameters are M˜ s = 5.34, M˜ a = 0.172, β˜ = 0.00108, δρ˜ δ˜B ρ = 1.37, B = 0.168 andσ ˜ = 0.56.

The behavior of the Alfvén wave pressure can also be tested numerically. We have chosen to take β = 0 (by turning off the thermal pressure term), in order to make its effect more visible. In a run with parallel propagation, the forcing is now applied on modes 49–51 for the perpendicular velocity components and on modes 1–5 for the longitudinal component. Three different simulations have been performed, with M˜ a = 0.073, 0.795 and 2.09. As visible in Figs. 11–13, while the magnetic pres- 1/2 sure roughly behaves as ρ for Ma = 0.4, a clear tendency to approach a polytropic behavior with γ = 2 is observed when Ma increases.

5. Conclusions 5.1. Summary and discussion In this paper we have investigated the dependence of the magnetic pressure, B2 (and therefore, of the magnetic field strength) on density in fully turbulent supersonic flows, parameterized by the Alfvénic Mach number Ma of the flow (which provides a more direct measure of the amount of field distorsion than the β parameter1) and the angle between the mean field direction and that of wave propagation. To do this, we investigated the properties of simple waves, paying particular attention to the relations between the fluctuations of the magnetic and velocity fields on one hand, and the density fluctuations on the other. 2 We then presented numerical experiments confirming the ex- Fig. 8. Top a) snapshot of log10 ρ (solid) and log10 b (dotted) for the run of Fig. 7a. Bottom b) density PDF for the run of| | Fig. 7a in log-log pectations from the analysis, and extended the results. coordinates. The dashed shows a log-normal fit. From the simple-wave analysis, we concluded that the production of density fluctuations is dominated by the slow mode at low Ma, while at large Ma both modes contribute. The fact of Alfvén wave pressure). In contrast with the case of large Ma, 1 For example, Lithwick & Goldreich (2001) considered high-β a larger scatter is observed on the magnetic pressure vs. den- plasmas, while Cho & Lazarian (2002) considered low-β ones. sity diagram, as seen in Fig. 9. As seen in Fig. 10a, the corre- Nevertheless, both groups restricted their studies to low-Ma regimes. lation between density and magnetic field intensity is lost and In the present work, instead, we have considered both small- the density PDF is closer to a log-normal (Fig. 10b). and large-Ma cases. T. Passot and E. Vázquez-Semadeni: Density-magnetic pressure correlation 853

Fig. 11. Log-log scatter plot of the magnetic pressure versus density for a simulation with θ = 0andβ = 0. The parameters are M˜ a = δρ˜ δ˜B 0.073, ρ = 3.94, B = 0.0564 andσ ˜ = 0.60. The line segment has a slope 0.5.

2 Fig. 10. Top a) snapshot of log10 ρ (solid) and log10 b (dotted) for the run of Fig. 9. Bottom b) density PDF for the run of| | Fig. 9 in log-log coordinates. The dashed shows a log-normal fit. that the fast mode can be neglected in accounting for the spec- trum of density fluctuations at Ma < 1 was already noted by Lithwick & Goldreich (2001). In this∼ paper we have shown that this result cannot be extrapolated to large values of Ma. Fig. 12. Log-log scatter plot of the magnetic pressure versus density The simple-wave analysis also showed that, except for the for a simulation with θ = 0andβ = 0 The parameters are M˜ a = 0.795, δρ˜ δ˜B case of small field fluctuations with simultaneously moderate ρ = 4.18, B = 0.311 andσ ˜ = 0.63. The line segment has a slope 2. density fluctuations and parallel field component, the magnetic pressure behaves as that turbulent pressure cannot be simply modeled with a poly- Pmag c1 c2ρ (39) ' − tropic law in general, and in fact it can act as a “random” forc- for the slow mode, and as ing instead of as a restoring force. In particular, strong density fluctuations are possible even in the presence of a large uniform P ρ2 (40) mag ' magnetic field, but in general there is no relation between δρ/ρ for the fast mode. This different scaling of the magnetic pres- and β˜. sure with density for the two modes is the basis of the decorre- The angle θ between the mean field and the direction of lation observed, as in general a given value of the density can wave propagation also plays an important role in determining be arrived at by a different kind of wave, and therefore have a the relative importance of the modes. At perpendicular propa- different associated value of the magnetic pressure. More gen- gation, the slow mode does not propagate, but at almost perpen- erally, we can say that the particular value of the magnetic pres- dicular propagation and low Ma it is dominant. At intermediate sure of a fluid parcel will not be uniquely determined by its angles, the distinction between the low- and high-Ma cases is density, but instead, that it will depend on the detailed history less pronounced, although at large Ma a tighter correlation be- of how the density fluctuation was arrived at. This also implies tween density and magnetic field strength is observed at high 854 T. Passot and E. Vázquez-Semadeni: Density-magnetic pressure correlation

other waves at intermediate angles. In this sense, it is likely that the most representative simulations of the full 3D case in this paper are those with intermediate values of the propagation angle θ, for which the density PDF is close to log-normal. Indeed, preliminary three-dimensional simulations in a variety of situations show that the density PDF is not very far from log-normal, as has also been found by V´azquez-Semadeni & Garc´ıa (2001) and Ostriker et al. (2001). Note, however, that in the simulations of the latter authors, a clear trend towards more extended high-density tails and a shift of the PDF maximum to low densities is observed at progressively lower values of β. Since those authors used ﬁxed sonic Mach numbers in all simulations, their β is a direct measure of Ma, and thus the trend they report shows that the increased importance of the slow mode at low Ma we have discussed is still present in 3D.

Fig. 13. Log-log scatter plot of the magnetic pressure versus density 5.2. Implications ˜ for a simulation with θ = 0andβ = 0 The parameters are Ma = 2.09, Our results have a number of interesting implications for δρ˜ δ˜B ρ = 3.503, B = 1.01 andσ ˜ = 0.76. The line segment has a slope 2. the standard picture for the support of molecular clouds by magnetic fields, and on the interpretation of observations. Observational surveys of the magnetic field strength in molec- densities. Also, in some cases (parallel propagation at moder- ular clouds seem to indicate that the former is essentially un- 3 3 ate to large Ma), density peaks can even correspond to mag- correlated from the density at low (<10 cm− ) densities, with netic pressure minima. This is in fact reminiscent of pressure- recorrelation seemingly appearing∼ at higher densities (e.g., balanced structures commonly observed in the solar wind and Crutcher et al. 2002). In particular, in molecular clouds non- possibly present in the ISM (see the simulations of Mac Low detections of the Zeeman effect are generally more frequent et al. 2001). than detections, and so in many cases only upper limits to the Our results have implications for the functional form of the field strength are available (e.g., Crutcher et al. 1993; Bourke density PDF. We found that it tends to be log-normal, due to et al. 2001). Although it has been argued that these results are the unsystematic action of the magnetic pressure, which al- consistent with the standard theory of molecular cloud support lows the thermal pressure to take control, except when the slow (Shu et al. 1987) through consideration of statistical correc- mode dominates density fluctuation production. In this case, tions to the random orientations of the field with respect to the strength of the random forcing-like action of the magnetic the line of sight (Crutcher et al. 1993; Crutcher et al. 2002), pressure becomes density-dependent, and causes a low-density it is clear that they are also consistent with the lack of corre- excess in the PDF. lation between field strength and density found in the present We also presented a perturbation analysis of the Alfvén- paper and in other numerical simulations (Padoan & Nordlund wave pressure, which recovered all the limiting polytropic 1999; Ostriker et al. 2001). This has been already pointed out cases obtained by MZ95, with γe 2atlargeMa, γe 3/2 by Padoan & Nordlund (1999). ' ' at moderate Ma,andγe 1/2atlowMa, but we also men- The recorrelation apparently observed at higher densities ' tioned a result by Passot & Gazol (in preparation) using this ap- (also noticeable in the 3D simulations of Ostriker et al. 2001 proach showing that the Alfvén wave pressure is in general not including self-gravity) is consistent with our cases with inter- isotropic. As pointed out by MZ95, the conclusion of isotropy mediate values of θ and large Ma, suggesting that the Alfvén follows from assuming a unique relation between the fluctua- Mach number in molecular clouds is relatively large. However, tions of the velocity and of the magnetic field, valid for a trav- it is also possible that this is an effect of self-gravity becoming eling Alfvén wave. That isotropy does not hold in general is important, which we have not considered here, or that, even evidence that all three wave modes contribute to the velocity at high densities, the magnetic field strength is highly variable fluctuations, each one with a different functional form, simi- from one core to another. More observations of the magnetic larlytothedifferent scalings of magnetic pressure with density field strength in high-density molecular cloud cores, reporting for the slow and fast modes. both detections and non detections, are needed. The question arises as to whether these results subsist in Acknowledgements. We thank the referee, Alex Lazarian, for a the three-dimensional situation, in which all waves and all di- prompt report and useful comments. This work has received partial rections of propagation co-exist. Without extensive numerical financial support from the French national program PCMI (France) simulations, it is difficult to predict what will be the relative to T. P. and from CONACYT (México) grant 27752-E to E. V.-S. weight of each process at play. It can be guessed that the strong signature of slow waves, seen in 1D for almost perpendicular References propagation and small values of Ma, will not be as important compared to the production of density fluctuations by all the Arons, J., & Max, C. E. 1975, ApJ, 196, L77 T. Passot and E. Vázquez-Semadeni: Density-magnetic pressure correlation 855

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