Downes, Turlough

O\u27Sullivan, Stephen

English

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Technological University Dublin ARROW@TU Dublin Conference papers School of Mathematics 2008-4 Implications of the Hall Effect for Turbulent Molecular Clouds Turlough Downes Dublin City University Stephen O'Sullivan Technological University Dublin, stephen.osullivan@tudublin.ie Follow this and additional works at: https://arrow.tudublin.ie/scschmatcon  Part of the Astrophysics and Astronomy Commons Recommended Citation Downes, T. and O'Sullivan, S. Implications of the Hall Effect for Turbulent Molecular Clouds. ASP Conference Series, Vol. 385, Copyright 2008. This Conference Paper is brought to you for free and open access by the School of Mathematics at ARROW@TU Dublin. It has been accepted for inclusion in Conference papers by an authorized administrator of ARROW@TU Dublin. For more information, please contact yvonne.desmond@tudublin.ie, arrow.admin@tudublin.ie, brian.widdis@tudublin.ie. This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License **FULL TITLE**ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION****NAMES OF EDITORS**Implications of the Hall effect for turbulent molecularcloudsTurlough P. DownesSchool of Mathematical Sciences, Dublin City University, Glasnevin,Dublin 9, IrelandStephen O’SullivanUCD School of Mathematical Sciences, University College Dublin,Belfield, Dublin 4, IrelandAbstract. We present the first study of turbulent decay in systems in whichthe Hall effect is important. We use a new, stable, explicit algorithm to integratethe multifluid MHD equations in the presence of the Hall effect.We find that in Hall-dominated systems effect introduces more structure indensity on all length scales, and marginally reduces the turbulent energy decayrate, than in equivalent systems which are ambipolar dominated.1. IntroductionTurbulence is believed to be an important factor in determining the evolution ofmolecular clouds, both in terms of their global behaviour (such as collapse) andalso in terms of the initial mass function of the stars formed in such clouds. Itis difficult to make progress analytically with studies of turbulence in a mediumsuch as a molecular cloud. Many authors (e.g. Mac Low et al. 1998; Mac Low1999; Ostriker et al. 2001; Vestuto et al. 2003; Gustaffson et al 2006; Glover &Mac Low 2007) have investigated the evolution of magnetohydrodynamic (MHD)turbulence, usually in the case of ideal MHD but sometimes also incorporatingambipolar diffusion, sometimes called “ion-neutral friction”.It is important to understand how turbulence in molecular clouds decaysas this provides a constraint on the injection of energy into the cloud, for ex-ample from star formation, to maintain the turbulence at observed levels. Forhydrodynamics it is well known that the kinetic energy in the fluid is expectedto decay as a power law of index − 107, while for compressible ideal MHD thesituation is somewhat less well understood the index for the decay appears tobe considerably lower than this (see e.g. Mac Low 1999 for a discussion),As noted by Wardle & Ng (1999) and Wardle (2004), the Hall effect can beimportant in parts of molecular clouds where the density is in the range 107 –1011 cm−3. In this paper we use a new numerical method to perform a prelimi-nary study of the effects on turbulence of the Hall effect and compare them withidentical simulations in which ambipolar diffusion is the dominant non-ideal ef-fect. The decay of turbulence in a single-fluid approximation incorporating thepresence of the Hall effect has been studied previously by Matthaeus et al. (2003)who found no significant effect on this quantity of the turbulence.12 Downes and O’SullivanIn this paper we perform low resolution simulations of turbulent decay in aweakly ionised multifluid MHD system as a means of gaining a first insight intothe behaviour of turbulence in such a system. The Hall effect, Ohmic diffusionand ambipolar diffusion are all present. We analyse the results in terms of thekinetic energy decay, the magnetic field generation/decay and the size of thestructures created in the density distribution by the turbulence.In section 2. we outline the equations used, in section 3. we give a briefoverview of the numerical algorithm employed. Section 4. gives the set-up forthe simulations, while we present the results themselves in section 5.. Finally,we present our conclusions in section 6..2. The multifluid MHD equationsThe equations describing multifluid MHD in the limit of low ionisation are (e.g.Falle 2003)∂ρi∂t+∇ · (ρivi) = 0, (1)∂ρ1v1∂t+∇ · (ρ1v1v1 + p1) = J×B, (2)∂B∂t+∇ · (v1B−Bv1) = −∇×E′ (3)αiρi (E + vi ×B) + ρiρ1Ki 1(v1 − vi) = 0, (4)∇ ·B = 0, (5)∇×B = J, (6)N∑i=2αiρi = 0,N∑i=2αiρiqi = J. (7)The subscripts denote the species, with a subscript of 1 indicating the neutralfluid. The symbols αi, ρi, vi and Pi represent the charge-to-mass ratio, the massdensity, velocity and pressure of species i. In the work presented here we assumean isothermal flow, although it is straightforward to include the more generalenergy equations in the above system. The motion of the various species arecoupled through the collisional coefficients Kij .The electric field in the neutral frame, E′ ≡ E + v1 ×B, can be written interms of the current, J asE′ = rO(J ·B)B2B + rHJ×BB+ rA(J×B)×BB2(8)where rO, rH and rA are the Ohmic, Hall and ambipolar resistivities respectively.For more details see O’Sullivan & Downes (2007).The Hall effect in molecular clouds 33. Numerical methodWe use the numerical method described in O’Sullivan & Downes (2006; 2007).The main extra challenge in integrating the multifluid MHD equations is indealing with the diffusion terms in the induction equation.Where ambipolar diffusion is important (i.e. when η ≡rArHis large) we canuse the super-time-stepping method for accelerating standard diffusion equa-tions (e.g. Alexiades et al. 1996). It is not, however, possible to use this methodfor the Hall term since the eigenvalues of the Hall diffusion operator are com-plex. Instead we use the Hall Diffusion Scheme which allows us to integratethis term explicitly with a reasonable time-step. This means that the methodis straightforward to implement in a parallel and/or adaptive mesh refinementenvironment. A more detailed description of this method is given in O’Sullivan& Downes (this volume).4. Initial ConditionsWe use initial conditions which are similar to those in Mac Low (1999). Thecomputational domain is initialised as a 643 box of side 2. The sound speed is0.1, so the crossing time is 20 units. The magnetic field is initially uniform inthe x direction. The density is also initially uniform with ρ = 1 everywhere.The full parameters for the simulations are given in tables 1 and 2.Each component of the velocity field is initialised to be a linear combina-tion of waves with 64 wave vectors (from k = (1, 1, 1)T to k = (4, 4, 4)T) eachwith a random amplitude and phase taken from a uniform distribution. Thefinal velocity field is then normalised to have the appropriate root-mean-squareamplitude (see table 2).Table 1. Parameters for the ambipolar and Hall dominated simulationsParameter Ambipolar dominated Hall dominatedSound speed 0.1 0.1ρ 1 1α2 −2× 1012−2× 109α3 1× 108 1× 105K12 4× 105 4× 102K13 2× 104 2.5× 106ρ2 5× 10−8 5× 10−8ρ3 1× 10−6 8× 10−12η 5.86× 103 4.6× 10−3For each of the sets of conditions in table 2 an ambipolar and a Hall domi-nated simulation was run (see table 1) With the parameter ranges chosen we areexploring three different regimes: weak magnetic fields with trans- and super-Alfve´nic turbulence initially, and a strong magnetic field case with trans-Alfve´nicturbulence.4 Downes and O’SullivanTable 2. Initial conditions for the simulations discussed in this work.Simulation B-field vrms β =PthermalPmagneticA B = (0.1, 0, 0)T 0.1 2B B = (0.1, 0, 0)T 0.2 2C B = (0.1, 0, 0)T 0.3 2D B = (0.1, 0, 0)T 0.4 2E B = (0.4, 0, 0)T 0.4 0.1255. ResultsWe analyse the results of the simulations now in terms of the decay of the kineticenergy in the flow, the evolution of the magnetic field and the length scales ofthe density structures which develop in the flow. We focus on comparing resultsobtained in the flows which are Hall dominated with those which are ambipolardominated.5.1. Energy decay 10 100 1000 10000 100000 0.1  1  10  100Integrated BTimev_rms = 0.4, ambipolar dominatedv_rms = 0.4, Hall dominatedv_rms = 0.1, ambipolar dominatedv_rms = 0.1, Hall dominated 1000 10000 0.1  1  10  100Integrated BTimev_rms = 0.4, ambipolar dominatedv_rms = 0.4, Hall dominatedv_rms = 0.1, ambipolar dominatedv_rms = 0.1, Hall dominatedFigure 1. Evolution of the total kinetic energy (left panel) and the inte-grated magnetic flux for simulations A and D.Table 3. Exponent of the kinetic energy decay with time.Simulation Hall dominated Ambipolar dominatedA 0.44 0.58B 0.68 0.76C 0.84 0.87D 0.93 0.98E 0.71 0.83Figure 1 contains plots of the decay of the total kinetic energy in the com-putational box as a function of time for simulations A and D. It is clear that,The Hall effect in molecular clouds 5after an initial transient, the decay does behave as a power law. Modelling thedecay with time as Ek ∝ t−γ , table 3 contains the values of γ obtained for eachof the simulations. It is noticeable that, in each case the Hall dominated simu-lation has a lower exponent, indicating that it maintains its kinetic energy forlonger than the ambipolar simulation.While this is a systematic effect in all our simulations, the differences aregenerally not very large. This may in part be due to the relatively low resolutionof the simulations presented in this work.5.2. B-field evolutionFigure 1 contains plots of the integrated magnetic flux in the computationaldomain as a function of time for simulations A and D. It is clear that in bothcases the magnetic field becomes more intense with time initially before dyingaway. In addition, simulation E develops more magnetic flux than simulationA. These two points can be explained as follows: as the flow evolves energyis transferred from the kinetic energy of the fluids into the magnetic field. Asthe kinetic energy in the motion of the fluids decays the magnetic energy istransferred back to the fluids, resulting in the decay of the magnetic field. If theinitial flow is more energetic (in the case of simulation E) then we expect thatmore magnetic energy would be generated initially.Finally we note that the ambipolar dominated flows do not generate quiteas much magnetic field as the equivalent Hall dominated flows. This, again, isto be expected since, for significant ambipolar diffusion, the flows do not haveas much “grip” on the magnetic fields so they do not generate the dynamoeffect necessary to increase the magnetic energy as efficiently. Since there is nodiffusion of the magnetic field in the usual sense for the Hall dominated flows,these flows can generate the magnetic energy more efficiently.5.3. Power spectraTable 4. Exponent of the power spectrum of density.Simulation Hall dominated Ambipolar dominatedA 6.73 6.65B 5.04 5.40C 4.20 4.66D 3.62 3.86E 5.69 6.67To gain an idea of the typical size of structure produced in each of the simu-lations we calculate power spectra of the density distribution in space, modellingthese spectra as P (k) ∝ k−γ . Table 4 contains the exponents calculated for eachsimulation. It can be seen that, systematically, the Hall dominated simulationshave harder spectra. Indeed, further analysis of the results (not shown here)show that these simulations have more power than the ambipolar dominatedsimulations at all scales.6 Downes and O’SullivanOne would expect that the Hall dominated flows should have more powerat shorter length scales than the ambipolar dominated flows for two reasons.Firstly the Hall effect is dispersive in nature. Hence large waves will, in general,break up into short wavelength waves travelling quickly and longer wavelengthwaves travelling more slowly, thereby generating structure on short scales. Inaddition, ambipolar diffusion, unlike the Hall effect, is a truly diffusive effect. Itwill tend to smear out structure on short scales.6. ConclusionsWe have presented the results of simulations of turbulence in molecular cloudsin the case where the flow is Hall dominated and compared these results withsimulations where the flow is ambipolar dominated. For each case we have runsimulations in which the initial velocity field is trans- and super-Alfve´nic, andin which β = 2 and β = 0.125.We find that the Hall effect produces more structure in the density dis-tribution at all length scales. It allows a greater transfer of energy from theflow of the fluids to the magnetic field than is the case for the ambipolar domi-nated case. We also find that the kinetic energy in Hall dominated flows decaysless rapidly than in equivalent ambipolar dominated flows, although this is asomewhat marginal effect in the simulations presented here, possibly due to therelatively low resolution employed.Acknowledgments. This work was partly funded by the CosmoGrid project,funded under the Programme for Research in Third Level Institutions (PRTLI)administered by the Irish Higher Education Authority under the National De-velopment Plan and with partial support from the European Regional Devel-opment Fund. The present work was also supported in part by the EuropeanCommunity’s Marie Curie Actions - Human Resource and Mobility within theJETSET (Jet Simulations, Experiments and Theory) network under contractMRTN-CT-2004 005592.ReferencesAlexiades V., Amiez G., Gremaud P., 1996, Com. Num. Meth. Eng., 12, 31Falle, S.A.E.G., 2003, MNRAS, 344, 1210Glover, S.C.O., Mac Low, M-M., 2007, ApJ, 659, 131Gustaffson, M., Brandenburg, A., Lemaire, J.L., Field, D., 2006, A&A, 454, 515Mac Low, M-M., 1999, ApJ, 524, 169Mac Low, M-M., Klessen, R.S., Burkert, A., Smith, M.D., 1998, Phys. Rev. Lett. 80,2754Matthaeus, W.H., Dmitruk, P., Smith, D., Ghosh, S., Oughton, S., 2003, Geophys. Res.Lett. 30(21), 2104Ostriker, E.C., Stone, J.M., Gammie, C.F., 2001, ApJ, 546, 980O’Sullivan, S., Downes, T.P., 2006, MNRAS, 366, 1329O’Sullivan, S., Downes, T.P., 2007, MNRAS, 376, 1648Vestuto, J.G., Ostriker, E.C., Stone, J.M., 2003, ApJ, 590, 858Wardle, M., Ng, C., 1999, MNRAS, 303, 239Wardle, M., 2004, Ap&SS, 292, 317

Implications of the Hall Effect for Turbulent Molecular Clouds

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Implications of the Hall Effect for Turbulent Molecular Clouds

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