The Impact of Magnetic Fields on Thermal Instability

MNRAS 476, 852–867 (2018) doi:10.1093/mnras/sty293 Advance Access publication 2018 February 5

The impact of magnetic ﬁelds on thermal instability

Suoqing Ji ( ),‹ S. Peng Oh and Michael McCourt† Department of Physics, University of California, Santa Barbara, CA 93106, USA

Accepted 2018 January 31. Received 2018 January 31; in original form 2017 September 29

ABSTRACT Downloaded from https://academic.oup.com/mnras/article/476/1/852/4839008 by guest on 28 September 2021 Cold (T ∼ 104 K) gas is very commonly found in both galactic and cluster haloes. There is no clear consensus on its origin. Such gas could be uplifted from the central galaxy by galactic or AGN winds. Alternatively, it could form in situ by thermal instability. Fragmentation into a multiphase medium has previously been shown in hydrodynamic simulations to take place once tcool/tff, the ratio of the cooling time to the free-fall time, falls below a threshold value. Here, we use 3D plane-parallel MHD simulations to investigate the influence of magnetic fields. We find that because magnetic tension suppresses buoyant oscillations of condensing cool ∼ gas, it destabilizes all scales below lA vAtcool, enhancing thermal instability. This effect is surprisingly independent of magnetic field orientation or cooling curve shape, and sets in even at very low magnetic field strengths. Magnetic fields critically modify both the amplitude and morphology of thermal instability, with δρ/ρ ∝ β−1/2, where β is the ratio of thermal to magnetic pressure. In galactic haloes, magnetic fields can render gas throughout the entire halo thermally unstable, and may be an attractive explanation for the ubiquity of cold gas, even in the haloes of passive, quenched galaxies. Key words: galaxies: clusters: general – galaxies: evolution – galaxies: haloes – galaxies: magnetic fields.

ing has found such widespread cold gas to be ubiquitous in all 1 INTRODUCTION surveyed quasars, out to distances ranging from ∼100 to 300 kpc. Modern theories of structure formation predict the haloes of galax- Understanding the origin of this cold gas is an important chal- ies and clusters to be filled with hot virialized gas in approximate lenge for several reasons. The CGM is an important discriminant of hydrostatic equilibrium (e.g. Birnboim & Dekel 2003). This is eas- galaxy formation models, which have largely been tuned to match ily seen directly in X-ray emission at group and cluster scales, and observed stellar masses and galaxy morphology. Furthermore, only at galaxy scales has been inferred from stacked SZ observations T ∼ 104–105 K gas will be observationally accessible in detail for (Planck Collaboration XI 2013; Anderson, Churazov & Bregman the foreseeable future. Finally, cold gas both provides fuel for star 2015) and more indirect measures such as confinement of gas clouds formation and must be expelled in outflows, and thus is arguably and stripping of satellite galaxies (Maller & Bullock 2004; Fang, the most important component to track and understand. Bullock & Boylan-Kolchin 2013;Stockeetal.2013). In recent The theoretical challenge is similar to that in galaxy cluster cores, years, a much greater surprise from quasar absorption line obser- where the observed cold filaments are thought to arise either from vations of the circumgalactic medium (CGM) was the ubiquity uplifted cold gas from the central cD galaxy, or are made in situ via and abundance of T ∼ 104 K, photoionized gas throughout such thermal instability. In galaxies, models where cold gas is uplifted by haloes, which is seen in more than half of sightlines within 100 kpc galactic winds face problems in understanding how gas can be en- in projection (Stocke et al. 2013;Werketal.2013). Photoioniza- trained without being shredded by hydrodynamic instabilities (e.g. tion modelling suggests reservoirs of cold gas of ∼1010–1011 M, Scannapieco & Bruggen¨ 2015; Zhang et al. 2017), leading to sug- which is ∼1–10 per cent of the halo mass (Stocke et al. 2013;Werk gestions that such gas must instead spawned by thermal instability et al. 2014; Stern et al. 2016). At higher redshifts, when quasars in the wind (Martin et al. 2015; Thompson et al. 2016; Scannapieco are prevalent, such cold gas is also seen in area-filling, smoothly 2017). It is also hard to understand the prevalence of dense cold distributed fluorescent Lyα emission (Cantalupo et al. 2014; gas in passive galaxies. These problems may potentially be allevi- Hennawi et al. 2015; Cai et al. 2017). Importantly, deep Lyα imag- ated in recent models which invoke small-scale structure in cold gas to enable rapid entrainment and suspension of cold gas droplets (McCourtetal.2018). None the less, thermal instability remains an attractive way to produce cold gas in situ and solve these problems. E-mail: [email protected] Simulations of local thermal instability in stratified systems in † Hubble Fellow. global hydrostatic and thermal balance showed that fragmentation

C 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society Impact of B-fields on thermal instability 853 into a multiphase medium happens if the ratio of the local cooling stability (Loewenstein 1990;Balbus1991). However, there are two time to the free-fall time falls below a threshold value of tcool/tff ∼ 1– important distinctions between this work and that of Loewenstein 10. Subsequent simulations by other groups have corroborated this (1990) and Balbus (1991). First, the background state they assume finding, and have also extended its relevance to another domain, is a cooling flow. In the purely hydrodynamic case, thermal insta- suggesting that black holes are fed by cold accretion, maintaining bility cannot develop (Balbus & Soker 1989;Li&Bryan2012). a feedback loop that pins tcool/tff at this threshold value (Gaspari, By contrast, we assume a background state in hydrodynamic and Ruszkowski & Sharma 2012; Gaspari, Ruszkowski & Oh 2013; thermal equilibrium, where thermal instability and a multiphase Li & Bryan 2014;Lietal.2015; Prasad, Sharma & Babul 2015). medium can develop, if tcool/tff is sufficiently low (McCourt et al. Observationally, there is also support for the idea that an interplay 2012;Sharmaetal.2012). Secondly, those papers perform a linear between cooling and feedback leads cluster cores to self-regulate at stability analysis. Since the damping of thermal instability by gas a threshold value tcool/tff ∼ 10 (Sharma et al. 2012; Voit & Donahue motions is fundamentally a non-linear process, it is only by run- 2015). The exact origin of this numerical value for the threshold ning MHD simulations that we can find the non-linear endstate of

is still debated (Voit et al. 2017), and the results of this paper will thermal instability. Downloaded from https://academic.oup.com/mnras/article/476/1/852/4839008 by guest on 28 September 2021 bear directly upon this. These ideas also lead to the notion that star The outline of this paper is as follows. In Section 2, we describe formation in galaxies may be precipitation regulated (Voit et al. our methods; in Section 3, we describe our numerical results; and in 2015). Section 4 we provide a physical interpretation. Finally, in Section 5 Observational constraints on magnetic fields in galaxy haloes we discuss the potential impact of some missing physics in our are relatively sparse and uncertain. In the Milky Way, best-fitting simulations, and summarize our conclusions. models indicate a field of 1–10 μG (see table 1 and section 17.4 of Haverkorn 2015). Assuming equipartition between cosmic rays and magnetic fields, the scale height from synchrotron emission is 2 METHODS 5–6 kpc (Cox 2005). Similarly large scale heights of ≥7 kpc are deduced1 from the radio haloes of edge-on galaxies (Dumke & We use FLASH v4.3 (Fryxell et al. 2000) developed by the FLASH Krause 1998; Heesen et al. 2009). Faraday rotation along radio Center of the University of Chicago for our simulations. A direction- quasar sightlines yield magnetic fields as large as tens of μG out to ally unsplit staggered mesh MHD solver, based on a finite-volume, ∼50 kpc distances from galaxies, out to z ∼ 1 (Bernet, Miniati & high-order Godunov method combined with a constrained trans- Lilly 2013), likely due to outflows. Faraday rotation also yields port type of scheme, is adopted to solve the following fundamen- ∼10 μG fields in the halo of the edge-on spiral NGC 4631 (Mora & tal governing equations of inviscid ideal magnetohydrodynamics Krause 2013).2 (Tzeferacos et al. 2012;Lee2013): At face value, these field strengths are far greater than the pres- ∂ρ + ∇ · (ρv) = 0, (1a) sure of gas in the cold phase (Werk et al. 2014), and comparable ∂t to or possibly higher than the pressure of the ambient hot gas3 (for which indirect measurements and upper limits suggest Phot 10−13 erg cm−3;Fangetal.2013; Anderson et al. 2015). This is in ∂ρv BB + ∇ · ρvv − + ∇P = ρ g, (1b) contrast to the previously studied situation in galaxy clusters, where ∂t 4π tot the magnetic field is highly subdominant: β = Pgas/Pmag ∼ 50. Such high magnetic fields may potentially arise from galactic outflows, ∂s the α– dynamo, or magnetic buoyancy of strong fields produced ρT + v ·∇s = H − L, ∂ (1c) in the galaxy. Regardless, observations suggest that the gas in the t outer parts of galaxy haloes is magnetically dominated, and therefore that the hydrodynamic calculations in the literature may miss ∂B + ∇ · (v B − Bv) = 0, (1d) essential aspects of the dynamics by neglecting magnetic fields. ∂t Indeed, strong magnetic fields can fundamentally change the physics of thermal instability. Local thermal instability in a stratified medium produces overstable gravity waves, whose non-linear satu- ∇·B = 0, (1e) ration depends on t /t , representing the competition between the cool ff = + 2 driving processes of cooling and the damping effect of buoyant os- where ptot p B /(8π) is the total pressure including both gas 2 cillations. Magnetic pressure can alter gas buoyancy, while magnetic pressure p and magnetic pressure B /(8π), g is gravitational ac- = −γ H L tension can suspend overdense gas. Our goal in this paper is to care- celeration, s cpln(Pρ ) is entropy per mass, and and are fully evaluate such effects in 3D MHD simulations. Previous work the source terms of heating and cooling. We adopt a plane-parallel = has already suggested that magnetic fields can promote thermal in- geometry with a symmetric domain about z 0 plane, where the gravitational acceleration aligns vertically along the z-axis:

z/a 1 Assuming equipartition, the total (ordered and irregular) B-field scale g = g0 z,ˆ (2) + 2 height is at least 3 − α times larger than the ∼2 kpc synchrotron scale height, 1 (z/a) ≈− where α 0.8 is the synchrotron spectral index (Beck & Wielebinski which describes a nearly constant gravitational acceleration for 2013). |z| > a with a smooth transition to zero at z = 0. Here, the soft- 2 For more details on magnetic field observations in galaxy haloes, see a ening distance is set to 1/10 of the disc scale height. The free-fall recent review paper Han (2017). 3 This could suggest that the cold phase is supported by non-thermal pres- time-scale associated with the gravitational field is sure such as magnetic fields, turbulence or cosmic rays (see Wiener, Oh & Zweibel 2017 for further discussion of the last possibility), and further high- 2z tff = , (3) light the importance of considering non-thermal components of the plasma. g0

MNRAS 476, 852–867 (2018) 854 S. Ji, S. P. Oh, and M. McCourt

With this gravitational acceleration we set up the initial condition in hydrostatic equilibrium, where temperature T0 is constant, the density profile ρ is a z2 ρ(z) = ρ exp − 1 + − 1 , (4) 0 H a2 and the scale height H = T0/g0. For convenience, we rescale our unit system by setting ρ0 = T0 = g0 = kB = μmp = 1. We performed all of our simulations in 3D Cartesian grids, with a resolution of 2563 for standard runs and higher resolutions for convergence tests. The domain of the simulation is set to be 6 × 6 × 6 in the units of scale height H. We adopt hydrostatic boundary conditions for the boundaries along z-direction to maintain hydrostatic Downloaded from https://academic.oup.com/mnras/article/476/1/852/4839008 by guest on 28 September 2021 equilibrium in the initial conditions when both cooling and heating are turned off, and periodic boundary conditions for other boundaries. We initialize isobaric density perturbations with a white noise power spectrum and amplitude of 1 per cent and wavenumber rang- Figure 1. Approximate power-law fits for cooling curve (dashed line), with ing from 1 to 20 in units of 2π/Lbox where Lbox is the domain size. index of −1 for galaxy haloes (blue) and 0.5 for galaxy clusters (orange). The magnetic fields are initialized in the horizontal direction with

β0 ranging from infinity to order unity. We also performed addi- maximum Tceiling (set to 5T0). Physically, these limits of Tcool and tional simulations with initially vertical fields, which are discussed Tceiling correspond to the turning points in cooling function at which in Section 4.2. These straight fields are initialized to have uniform gas can reach at the two-phase semisteady state. strength throughout the box, so in the initial background state the The cooling time in multiphase gas can become very much shorter magnetic fields do not exert any forces (and do not contribute to than the dynamical time. Computational methods which limit the hydrostatic equilibrium). Note that our setup implies that plasma time-step to some fraction of a cooling time thus become pro- beta falls with height in the background state. The value we quote hibitively expensive. We therefore implement the ‘exact’ cooling is β evaluated at z = H. algorithm described in Townsend (2009)intoFLASH code, where we It is well known that halo gas in galaxies and galaxy clusters resolve the operator-split energy equation quires some form of ‘feedback heating’ to prevent a global cooling dT (T ) catastrophe. Precisely how feedback works is still an active area of tcool =−Tref (7) research. Hence, we will follow McCourt et al. (2012) and imple- dt (Tref ) ment a simple model which preserves thermal energy at each radius by separating variables where Tref is arbitrary reference temperature, (or z) in the domain: to maintain global thermal balance, in each which has an analytic solution for (piecewise) power-law cooling time-step we sum up total amount of cooling over the volume of a functions. With this implementation, we can simulate rapid cooling, thin slab perpendicular to z-axis, and distribute heat uniformly over even when the cooling time dips below the simulation time-step. this volume. In the region of |z|≤a, heating and cooling are turned off due to the uncertainty of feedback prescription at small radii. 3 RESULTS This idealization allows us to focus on the physics of thermal instability. Later works (e.g. Sharma et al. 2012; Gaspari et al. 2013)have We perform a series of simulations with different initial values of shown that results are not sensitive to the specific implementation tcool/tff and magnetic field strengths, and we plot the magnitude of heating. of density fluctuations as a function of tcool/tff in Fig. 2.Inthis As an approximate fit to slopes in the regimes of heavy element figure, each data point is taken from an individual simulation with excitation and bremsstrahlung respectively (Fig. 1), we implement the corresponding value of initial tcool/tff. The magnitude of den- cluster-like and galaxy-like power-law cooling functions as the fol- sity fluctuations is computed by volumetrically averaging over the lowing: region 0.9H < |z| < 1.1H.Itisdefinedas 2 1/2 − δρ ρ − ρ L = 2 = 0n T (T Tfloor) galaxy cluster = n (T ) 2 −1 , (5) , (8) 0n T (T − Tfloor) galaxy halo ρ ρ where is Heaviside function. These cooling functions correspond where the bracket represents the spatial averaging over at the same to two cooling time-scales: height. This quantity is computed from a snapshot of each simula- ⎧ ∼ 1/2 tion which has been evolved over 17 tcool, when thermal instability ⎨ 5 T galaxy cluster 3 n 0 saturates and the computed quantity enters a stable stage. Note that tcool = . (6) ⎩ 2 in our unit system we change the cooling time by changing ; 5 T galaxy halo 0 6 n 0 physically, however, this corresponds to changing the gas density. The different prefactors are a result of the different cooling curve slopes (see equations 8 and 9 in Sharma et al. 2012). The T (set to floor 3.1 Amplitude of density fluctuations T0/20) limit prevents cold gas from collapsing to very small values which we cannot resolve. Physically, our assumption is that once In Fig. 2,weshowδρ/ρ as a function of tcool/tff for various val- gas reaches this floor, its cooling time becomes so short that it will ues of magnetic field strength and a galaxy cluster-like cooling cool all the way to T ∼ 104 K. In addition, the temperature could curve ( ∝ T−1/2). The hydrodynamic simulations have a δρ/ρ ∝ −1 reach arbitrarily large values under our simple heating prescription, (tcool/tff) scaling, as found by McCourt et al. (2012). Two strik- which is not physical. We therefore limit the temperature to some ing facts emerge from this plot. For this cooling curve, the MHD

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Figure 2. δρ/ρ as a function of tcool/tff with galaxy cluster cooling curve and different initial strengths of magnetic field characterized by the values Figure 4. The mass fraction of cold gas (defined as T < T0/3) over entire of β :blue–β =∞, orange – β = 772, green – β = 278, red – β = 27, volume, with galaxy cluster cooling curve and different initial strengths of 0 0 0 0 0 =∞ purple – β = 4, brown – β = 3. The dashed lines are power-law fits magnetic field characterized by the values of β0:blue–β0 , orange 0 0 = = = = to simulation data points. The dot markers are simulations with initially – β0 772, green – β0 278, red – β0 27, purple – β0 4, brown – = horizontal field, star markers are with vertical field. All the quantities are β0 3. The solid lines are with cluster cooling curve, and the dashed ones =∞ = measured at z = H. are with galaxy cooling curve. For β0 and β0 772 cases, there is no detectable cold gas for simulations with tcool/tff ≥ 2.8.

cooling curve we find that the simulations reach a steady-state amplitude: ⎧ −1 ⎪ tcool ⎨⎪0.1 hydro δρ t ∼ ff . (9) ρ ⎪ −1 ⎩⎪ −1/2 tcool 0.3β100 MHD tff While these fits are derived for horizontal fields and a cluster-like cooling curve, we shall see that (as hinted in Fig. 3) they are independent of cooling curve slope and field geometry down to β a few. The amplitude of density fluctuations is important because it signals whether spatially extended multiphase structure can develop in the saturated state of thermal instability. In Fig. 4, we show the cold mass fraction (where T < T0/3) as a function of tcool/tff.For stronger magnetic fields, the threshold value of tcool/tff at which Figure 3. The β dependence of δρ/ρ, where both δρ/ρ and β are taken multiphase gas can form is higher by up to an order of magnitude. from the snapshots at tcool/tff = 5.7 and averaged by volume over a thin Since tcool/tff varies as a function of radius, this implies that cold slab at z = H. The blue dots are with galaxy cluster cooling curve, and the gas can be produced over an increasingly wider radial range for orange dots are with the galaxy halo cooling curve. magnetized halo gas.5 In Fig. 4, the cold gas fraction apparently does not scale monotonically with β at high β (note the cross-over at tcool/tff = 0.7 between −1 = = simulations retain the overall scaling δρ/ρ ∝ (tcool/tff) , albeit the β 278 (green), β 772, and hydro (blue) curves, in apparent − with a normalization which increases as the magnetic field strength contradiction with the monotonic scaling of δρ/ρ ∝ β 1/2 seen in increases (i.e. plasma β decreases). Evidently, magnetic fields pro- Fig. 3. In fact, at higher tcool/tff (off the plot scale), there is more mote thermal instability. In Fig. 3, we quantify this:4 δρ/ρ ∝ β−1/2 cold gas in the β = 278 simulation than in the β = 772 and hydro for both galaxy and cluster-like cooling curves, with a strong uptick simulation, where cold gas is undetectable. This is a consequence signalling runaway cooling at low β for galaxy-like cooling curves. of the somewhat arbitrary definition of cold gas as T < T0/3. The Secondly, magnetic fields have an impact even when they are re- gas density PDFs narrow monotonically as β increases (Fig. 5), markably weak. Fig. 2 shows that the MHD case converges to the which is the more physically important measure. The cold gas mass hydrodynamic case only when β ∼ 1000. In short, for a cluster becomes sensitive to the exact definition as the PDF narrows. Also, note that while overall trends should be robust, detailed results are sensitive to both the cooling curve shape (which we have simplified)

4 5 We only plot this scaling for tcool/tff = 5.7, but it continues to hold for Strikingly, the cold fraction becomes independent of tcool/tff for low β and other values of tcool/tff. a galaxy cooling curve, an effect we soon discuss.

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weak initial fields, β correlates strongly with overdensity and can drop by a factor of ∼10 or more in regions within high overdensities; thus, magnetic fields can have a more significant dynamical effect than the mean or initial β0 might indicate. Furthermore, as discussed in Section 4 [see equations (22) and (23), and Fig. 12], what matters is not β = P/PB but δP/PB ∼ O(1). The magnetic stresses should not be compared against pressure forces (which are nearly balanced by gravity), but with the perturbed pressure forces, with which they are comparable. In the hydrodynamic case, Fig. 6(a), density fluctuations grow at the largest available scales, and show clear horizontal stratification. This horizontal stratification is characteristic of low-frequency g

modes, where buoyancy forces inhibit vertical motions. By contrast, Downloaded from https://academic.oup.com/mnras/article/476/1/852/4839008 by guest on 28 September 2021 once magnetic fields are introduced, the morphology of density fluctuations changes completely. For very weak initial fields, for example with β0 = 278 (Fig. 6b), cold gas is vertically oriented and has much more small-scale structure. As the magnetic field strength increases (β = 27, 3, Figs 6c and d), the vertical bias persists, but Figure 5. The probability density distribution of gas density measured at 0 the density fluctuations appear on increasingly larger scales. Finally, z = H, with galaxy cluster cooling curve and different initial strengths of if we re-run the simulation with strong magnetic fields β0 = 3, magnetic field characterized by the values of β0:blue–β0 =∞, orange t t = – β0 = 772, green – β0 = 278, red – β0 = 27, purple – β0 = 4, brown – but shorten the cooling time by a factor of 20 ( cool/ ff 0.28, β0 = 3. Simulations used here are with cluster cooling curve and at long Fig. 6e), then vertically oriented overdense filaments with much cooling time (tcool/tff = 5.7). Also shown with a dashed line is the β0 = 3 more small-scale structure appear. Later, we shall argue that these galaxy cooling curve. variations arise because the maximally unstable modes scale as ∼vAtcool; thus, weaker magnetic fields or shorter cooling times result in more structure at smaller scales. Eventually, as β ∼ 1000, the and also our cooling curve cutoff at T0/20. In reality, of course, the system converges to the hydrodynamic case, as we discuss further density PDF should be bimodal. in Section 4. To reinforce this point that magnetic fields can play a key role in the development of a multiphase medium, in Fig. 5 we show the PDF of gas density for different values of β, for the same parame- 3.3 Impact of field orientation ters (z = H, cluster cooling curve, tcool/tff = 5.7). Stronger magnetic fields (decreasing β) clearly broaden the gas density considerably. Clearly, magnetic fields change the structure of g modes seen in This is reminiscent of how decreasing values of tcool/tff broaden the the hydrodynamic case. Since magnetic fields introduce anisotropic gas density PDF (McCourt et al. 2012), and buttresses the expecta- stresses, one might expect their impact to change with magnetic tion from Fig. 2 and equation (9) that higher values of tcool/tff can be field orientation. For example, one might reasonably expect that in compensated with stronger magnetic fields to produce comparable the horizontal field case, magnetic tension supports overdense gas density fluctuations, even if (as we shall see) the morphology of and suppresses buoyancy-driven g modes. This effect should vanish cold gas is completely different. for vertically oriented fields. Indeed, when we run simulations with the same initial plasma β0 and tcool/tff = 5.7 but vertical fields, the morphology of density fluctuations appears completely different = 3.2 Morphology (Fig. 7). For β0 278 (Fig. 7a, compare with Fig. 6b), the density fluctuations appear larger scale and more planar. Note that the mag- In Figs 6(a)–(d), we show slice plots of the magnitude of den- netic fields become more horizontal in overdense regions; there is sity fluctuations for simulations utilizing the cluster cooling curve, some degree of magnetic tension support. By contrast, for β0 = 3 for tcool/tff = 5.7 at z = 1 (i.e. at one scale height) and initial (Fig. 7b, compare with Fig. 6d), density fluctuations appear at very β0 =∞(hydro), 278, 27, 3 corresponding to RMS density fluctua- small scales and manifest as thin vertical strips of alternating high −2 −2 tions of δρ/ρ RMS = 2.5 × 10 ,9× 10 , 0.15, 0.27. Note that and low density. We shall see that here, magnetic tension support is since the dynamic range of δρ/ρ RMS is different for each plot, the even more important (see Fig. 12d). colorbar scale is also different for each plot. Also superimposed are Incredibly, despite the strikingly different gas morphologies in the magnetic field lines. While the mean β in the entire box does simulations with initially horizontally and vertically aligned mag- not evolve significantly, it is clear that magnetic fields are amplified netic fields, the final RMS density fluctuations are remarkably in- around overdense regions. The origin of this amplification can be dependent of field orientation, and depend only on β and tcool/tff. deduced by evaluating the stretching and compressional terms in In Fig. 2, we indicate with stars the result of a few different simula- the flux freezing equation (Ji et al. 2016): tions with vertical fields. We see that in every instance δρ/ρ RMS closely corresponds to that derived from the equivalent horizontal stretching compression field simulations with the same β0 and tcool/tff. This remarkable 1 d|B|2 1 2 = B · (B ·∇)v − |B|2∇ · v − |B|2∇ · v . (10) result suggests that the total amount of cold gas is insensitive to 2 dt 3 3 the (in general unknown) field orientation. This is extremely sur- For high β, the effects of stretching and compression are com- prising, given how strongly the appearance of density fluctuations parable, while in the low β case, when magnetic tension is strong, varies with field orientation; we return to this result in Section 4, amplification is weaker and mostly compressional. In particular, for below.

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Figure 6. Slice plots of the magnitude of density ﬂuctuations with different initial β0 values and cooling curves, superposed with magnetic ﬁeld lines.

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Figure 7. Slice plots of the magnitude of density fluctuations with tcool/tff = 5.7 and initially vertical magnetic fields, superposed with line integral convolution of magnetic vector field.

tion, gravity drops out of the problem and a multiphase medium will always form, independent of tcool/tff. Importantly, this only happens for horizontal fields: the vertical field results show the same δρ/ρ −1 ∝ (tcool/tff) scaling at both high and low β, similar to the cluster cooling curve case. We show in Section 4, however, that closer examination reveals this to be actually a box size effect. In Fig. 8, we also show with a star the result of a simulation with the same effective resolution but double the vertical extent (6H × 6H × 12H). The density fluctuation has fallen considerably, and is close to the −1 (tcool/tff) scaling still obeyed by the cluster cooling curve. Exam- ination of the slice plots for the smaller and larger boxes show that both only have one dominant filament, with vAtcool > Lbox in the smaller box, suggesting that box size effects (Section 3.6) might play a role. We will return to this point in Section 4.3. The slice plots of the density field for β0 =∞, 278, 27 for the galaxy cooling curve are similar to that for the cluster case and are not shown. More interesting are the slice plots for low β.Itisinter- Figure 8. With galaxy cooling curve, δρ/ρ as a function of tcool/tff with esting to contrast the galaxy cooling curve case, Fig. 6(f) (β0 = 3, different initial strengths of magnetic field characterized by the values of tcool/tff = 5.7, δρ/ρ = 2.7760) with the cluster cooling curve case, =∞ = = = β0:blue–β0 , green – β0 278, red – β0 27, purple – β0 4, Fig. 6(d) (β = 3, t /t = 5.7, δρ/ρ = 0.2676). Both of these = 0 cool ff brown – β0 3. The dot markers are simulations with initially horizontal slice plots slow the cold gas to be condensed in only ∼2 vertically field and star markers vertical field. oriented structures in the entire simulation volume (one each above and below the mid-plane). This is in contrast to the β0 = 278, 27 3.4 Dependence on cooling curve cases (Figs 6b and c), where multiple such structures are visible. However, the cold gas in the galaxy cooling curve case reaches much In Fig. 8,weshowδρ/ρ as a function of tcool/tff for the galaxy-like higher peak overdensities, and thus has a much more pronounced ∝ −1 =∞ cooling curve T . For high β (β0 , 278, 27), the amplitude filamentary structure. Finally, the contrast between Figs 6(f) and of the density fluctuations appears to be roughly similar to the clus- (e) (β0 = 3, tcool/tff = 0.28) is instructive. Although both have ∝ −1 −1/2 ter cooling curve case, with δρ/ρ (tcool/tff) β independent similar RMS density fluctuations and cold gas masses, they have = of magnetic field orientation. However, at low β (β0 3, 4) and very different topologies. Fig. 6(e) shows much more small-scale for horizontal fields, density fluctuations appear to saturate with a fragmentation and appears more similar to Fig. 6(b), albeit with a 6 non-linear amplitude of δρ/ρ > 1, independent of tcool/tff. At face higher density contrast. We will discuss this in Section 4 in light of value, this result implies when magnetic fields approach equiparti- the characteristic scale vAtcool. Except at low β (discussed further in Section 4.3), these results 6 ∼ are consistent with a picture where the maximally unstable length The saturation√ amplitude δρ/ρ 3 is close to the maximum allowed ∼ δρ/ρ ∼ δ − 1 ∼ 4.4 as the mass fraction in cold gas with overdensity scale (which we shall later identify as vAtcool) depends only on β and t /t , independent of the shape of the cooling curve. δ ∼ T/Tfloor ∼ 20 approaches unity. cool ff

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Figure 9. Convergence tests. Panel (a): domain size of H × H × 2H, and resolutions of 64 × 64 × 128, 128 × 128 × 256 and 256 × 256 × 512 from left to right columns, at tcool/tff = 5.7. Panel (b): domain size of 6H × 6H × 3H, and resolutions of 256 × 256 × 128 and 512 × 512 × 256 from left to right columns, at tcool/tff = 5.7.

3.5 Convergence; 2D versus 3D show the results of simulations for β0 = 278 with ∼64, 128, 256 cells per scale height H (or ∼13, 27, 54 cells per lcool). Note that We show the results of convergence tests on a smaller box in Fig. 9. A unlike our fiducial simulations, here we only simulate the upper Our fiducial simulations are of size 6H (where H is the scale height) half (z > 0) of the box. We see that while the detailed morphology and 2563, and thus have ∼42 cells per scale height. It is also useful of the slice plots changes (with more small-scale detail apparent to phrase resolution in terms of the length scales: at higher resolution), the amplitude of density fluctuations is stable t ∼ cool ∼ ∼ cool at the 10–20 per cent level (and similar to that in our fiducial lhydro cstcool H , (11) = tff box, δρ/ρ 0.0899). We shall argue that this is because density cool fluctuations peak at the scale lA , which is well resolved. Similarly, in Fig. 9(b), we show the results of simulations for β = 3, with ∼42, H t 0 lcool ∼ v t ∼ cool , (12) 85 cells per H (or ∼21, 42 cells per lcool), which also show converged A A cool β1/2 t A ff values for δρ/ρ despite differences in detailed morphology. Note 2 × × where we have used H ∼ cstff and β ∼ (cs/vA) . We discuss the the differing box size for these convergence tests (H H 2H in relevance of these length scales further in Section 4.1. While these Fig. 9a, 6H × 6H × 3H in Fig. 9b). As is apparent in Fig. 9(b), this length scales are generally well resolved, in some regions of param- change is necessary because of the much larger unstable modes in eter space where tcool/tff < 1, β 1, they are not. This applies to the low β case. We will explore this further in the following section. data points in the short cooling time limit at high β in Fig. 2,where We have also run several 2D simulations and compared them to our simulations may be less reliable. our 3D results. In general, we find that 2D simulations produce In Fig. 9, we present the results of convergence studies for both higher density fluctuations by a factor of 2–3. 2D simulations were high (β0 = 278) and low (β0 = 3) β for a fixed value of tcool/tff = 5.7 already seen to produce somewhat higher density fluctuations in with horizontal fields and a cluster-like cooling curve. Tests in other the hydrodynamic simulations of McCourt et al. (2012); the effect regions of parameter space give similar results. In Fig. 9(a), we is more significant in MHD simulations. The reduced degrees of

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Figure 10. Overcooling effect: simulations of tcool/tff = 5.7 and β0 = 3 utilizing cluster cooling curve, with the same effective resolution of 64 cells per H, but different box widths of 8 × 4 × 2, 4 × 4 × 2and1× 1 × 2. freedom constrains gas motions, particularly in the anisotropic changes with both plasma β and tcool/tff.Wenowprovideaphysical MHD case, resulting in reduced damping of thermal instabilities. interpretation of these results. For this problem, it is important to use 3D simulations.

4.1 A characteristic scale 3.6 Effect of box size In order to understand the preceding results, it is useful to consider the dispersion relation for a magneto-gravity wave (Stein & In Fig. 10, we show the slice plots from simulations of different box × × × × × × Leibacher 1974): size (8 4 2, 4 4 2, and 1 1 2) but with the same effective ∼ ∼ cool = 2 resolution ( 64 cells per H,or 32 cells per lA ), for tcool/tff 5.7, k⊥ = ω2 = ω2 + k2 v2 , (13) β0 3 and the cluster cooling curve. From the preceding section, k BV B A this should produce converged RMS density fluctuations. Indeed, 2 ∼ ∼ −2 the δρ/ρ for the largest box is within ∼20 per cent of that of our where ωBV g/H tff is the Brunt–Vais¨ al¨ a¨ frequency, k⊥ in- fiducial simulation. However, for smaller boxes at fixed effective dicates transverse wave numbers perpendicular to the direction of ˆ resolutions, δρ/ρ increases monotonically, and is double the fidu- gravity, and kB ≡ k · b0 is the wavenumber parallel to the back- cial value in the smallest box! This is despite the fact that the slice ground field direction. The two terms on the right-hand side rep- plots appear broadly similar, with ∼2 large scale filaments per box. resent g mode and Alfven-wave´ oscillations, where the restoring cool Later, we shall argue that this implies that lA /Lbox is an important forces are buoyancy forces and magnetic tension, respectively. At cool dimensionless parameter: we require lA

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We can estimate the magnitude of density ﬂuctuations in the From these considerations, we can understand how magnetic hydrodynamic case as follows (McCourt et al. 2012). The linearized stresses change the amplitude of density ﬂuctuations. Tension forces momentum equation balance gravity when

2 dv ∼ B ρ = δρg (14) δρg , (19) dt λB where the magnetic tension force is (B ·∇)B = nBˆ 2/λ ,where gives a characteristic velocity of B the unit vector nˆ points to the centre of the radius of curvature, and δρ δρ H λB is the radius of curvature. Note that this is a non-linear criteria, v ∼ gtbuoy ∼ , (15) ∼ ∼ cool where δB B. Setting λB lA , the maximal scale which can ρ ρ tff be influenced by magnetic tension during thermal instability, we where we have used t ∼ t for the buoyant oscillation period obtain buoy ff ∼ 2 −1 Downloaded from https://academic.oup.com/mnras/article/476/1/852/4839008 by guest on 28 September 2021 and g H/tff ,whereH is the pressure scale height. The condition δρ t ∼ cool −1/2 tdrive ∼ tdamp yields the fastest growing perturbations if we maximize A2 β , (20) ρ tff tdamp ∼ L/v, setting L ∼ H. Setting tdrive ∼ tcool, we thus obtain which is consistent with our numerical results (equation 9; Figs 2 −1 and 3)whichshowδρ/ρ ∝ β−1/2. δρ ∼ tcool A1 (16) Comparing equations (16) and (20), we see that the MHD density ρ tff fluctuations approaches the hydrodynamic case when in good agreement with our numerical results (equation 9; Fig. 3), 2 A where in planar geometry the dimensionless coefficient A ∼ 0.1. β ∼ 2 ∼ 900, (21) 1 A Equation (14) is clearly modified in MHD case, when magnetic 1 stresses have to be considered. For instance, with an initially hor- where we have taken A1, A2 from equation (9). From Fig. 2,wesee izontal field, magnetic tension opposes gravity, and an overdense that this is indeed the case. Thus, magnetic stresses start to become fluid element can remain static. This suppression of buoyant oscilla- important even when the magnetic energy is highly subdominant. cool ∼ cool tions can be shown formally in a linear stability analysis, resulting in The characteristic scale lA vAt also explains why large cooling rates essentially identical to Field-type condensation modes boxes are needed for converged results in the low β, long cooling cool in an unstratified medium (Loewenstein 1990;Balbus1991). How- time limit. If the box is smaller than lA , then all modes within the ever, this effect is clearly scale dependent. Magnetic stresses can box are destabilized by magnetic fields (for instance, Fig. 10c, where cool only be significant during thermal instability if the Alfven´ crossing Lbox < 0.5lA ); gas motions are mostly strongly quenched for these time across a perturbation is less than the cooling time l/vA < tcool, small-scale modes. Since damping of gas motions is artificially which implies that only length scales: boosted, density fluctuations are larger for small boxes. Only when cool the largest unstable mode lA is contained within the box do the cool ≡ l

cool Our simulations show that the length scale lA (equation 17) is the relevant one, not l (contrast Figs 6d and e, where l is the same in 7 2 A A In a stably stratiﬁed medium, v2/v1 ∼ (v/ωBVH) 1 for weak tur- cool both simulations since β is the same, but lA is smaller by a factor bulence (e.g. see section 2 of Ruszkowski & Oh 2010). Note that strong of ∼20 in Fig. 6(e), due to the smaller value of tcool/tff.Thelatter turbulence can overwhelm buoyant restoring forces and thereby promote clearly shows much more small-scale structure). thermal instability (Voit, in preparation).

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Figure 11. Selected particle temporal trajectories coloured by entropy (left-hand panel) and spatial trajectories (right-hand panel, projected to x–z plane and horizontally shifted to x = 0 for starting point). The criterion for particle selection is the maximum entropy decrease from t = 0tot ∼ 17 tcool (dotted vertical line) at different heights, which helps to trace the pathway of cold gas formation. All the cases presented above are at long cooling time (tcool/tff = 5.7). is increased, buoyant oscillations are more strongly damped, and morphology of overdense filaments (Fig. 6b). By contrast, when = cool particles cool earlier at higher z,wheretcool/tff is larger (compare fields are strong (β0 3), lA is large, and particles can travel a left-hand panels of Figs 11b and c). When the field is weak, the considerable horizontal distance as they cool (right-hand panel of cool ∼ characteristic length scale lA vAtcool is small. Particles only Fig. 11c), resulting in filaments which are not necessarily vertically travel a small horizontal distance in a cooling time before they can- oriented (Fig. 6d). not be supported by the magnetic field and fall (right-hand panel We can also directly verify the importance of magnetic ten- of Fig. 11b). This accounts for the small scale, vertically oriented sion in supporting overdense filaments, by examining terms in the

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Figure 11 – continued linearized, perturbed Euler equation. In Fig. 12(a), we compare In Figs 11(d) and (e), we show particle trajectories for the the gravitational force on an overdense filament with the perturbed β0 = 278, 3 cases with vertical fields and tcool/tff = 5.7. In both pressure gradient ∂zδPtot (where Ptot = Pg + PB) and the perturbed cases, buoyant oscillations are suppressed, with amplitudes com- magnetic tension forces (B ·∇)Bz/4π, for the β0 = 278, horizon- parable to the horizontal field case (compare with Figs 11bandc). tal field case. We note that the magnetic tension term corresponds However, the spatial anisotropy of motions appears to be reversed: closely in morphology to the δρg term, providing most ∼70 per cent motions are horizontally (vertically) biased for the weak (strong) of the support against gravity. On the other hand, pressure gradients field case respectively, opposite to what is seen in the horizontal contribute a non-negligible amount (∼30 per cent) and can reach field case. This squares with the horizontal (vertical) orientation of comparable peak amplitudes, although the spatial correspondence overdense filaments in the weak (strong) vertical field case (Fig. 7), is less clear, and their time averaged influence is less important. compared to the opposite orientation in the horizontal field case cool ∼ Thus, the existence of a characteristic length scale lA vAtcool (Figs 6b and d). can explain three interesting features seen in our simulations: the Why are buoyant oscillations suppressed, even for vertical fields? dependence of filament size with β and tcool/tff; the existence of a Consider an overdense blob falling under the influence of gravity. minimum box size for converged results; and mostly importantly, Ram pressure leads to a high-pressure region upstream of the blob. the scaling of RMS density fluctuations with β. In the hydrodynamic (and horizontal field) case, this high-pressure region is dispersed by lateral gas flows, where unlike in the vertical direction, there are no restoring forces. However, in the vertical 4.2 Impact of magnetic field orientation field case, horizontal motions are restrained by magnetic tension In the preceding subsection, we made the case that magnetic field forces. The high-pressure region expands, bending the field lines tension damps buoyant oscillations by allowing for magnetic sup- until the horizontal pressure gradients balance magnetic tension port of overdense gas. Naively, one might expect that this effect will forces. We have verified this force balance explicitly in our sim- vanish for vertical fields, and thermal instability will no longer be ulations. However, this now implies (i) vertical pressure gradients enhanced. Instead, we saw that the amplitude of density fluctuations (ii) bent field lines offset from vertical which can exert magnetic was roughly independent of field orientation, at fixed β and tcool/tff. tension forces. Both of these can support overdense blobs against Whyisthisso? gravity and thereby suppress buoyant oscillations.

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cool ∼ 8 We can write the vertical and horizontal force balance equations lA vAtcool >Lbox. The density and entropy PDFs become bias modal, indicating genuine fragmentation into a multiphase medium. By contrast, in the larger box with identical t /t and β, the hot (B ·∇)Bz cool ff ∂ + = cool zδPtot δρg (22) medium does not evolve signiﬁcantly, with l

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Figure 12. Slice plots of the terms of overdensity gravity, total pressure gradient, and magnetic tension.

MNRAS 476, 852–867 (2018) 866 S. Ji, S. P. Oh, and M. McCourt with field-aligned conduction.9 Note that the suppression factor f density perturbations are large, but this regime has not been ex- is unknown and potentially large; there is no evidence for thermal plored in our study. Therefore, it is necessary in future studies to conduction anywhere in dilute plasmas apart from the solar wind, relax the assumptions about hydrostatic equilibrium by including which has a large scale, ordered magnetic field (Bale et al. 2013). driven turbulence into simulations, and investigate how initially (ii) Geometry. We have only simulated planar geometry, and the large density perturbations produced from strong turbulence can cool impact of spherical geometry could be important, once λA be- modify the evolution of thermal instability in MHD cases. cool ∼ comes large and approaches the radius r. Indeed, once λA /r cool ∼ −1/2 λA /H (tcool/tff )β > 1, we might expect magnetic fields to destabilize all available modes, similar to the ‘overcooling effect’ 5.2 Conclusions seen in small boxes with planar geometry (see Section 3.6). This In this paper, we explore the effects of magnetic fields on thermal will have to be explored in future work. Note that in the hydro- instability in a stratified medium with 3D MHD simulations. Astro- dynamic case, spherical geometry was previously thought to boost physical plasmas are invariably magnetized and our main message the threshold value of tcool/tff required for a multiphase medium is that this is a regime where pure hydro is clearly insufficient; MHD Downloaded from https://academic.oup.com/mnras/article/476/1/852/4839008 by guest on 28 September 2021 to develop due to differing compression on a sinking fluid element effects simply cannot be swept under the rug. The main application (Sharma et al. 2012), but it has subsequently been shown that once we have in mind is the formation of cold gas in galaxy (β ∼ few) an apples-to-apples comparison of where the cold gas appears is and cluster (β ∼ 50) haloes, but these results should be broadly performed, there is little to no difference between spherical and applicable to other environments. We find that magnetic fields have Cartesian geometry (Choudhury & Sharma 2016). The latter au- a striking impact on both the amplitude and morphology of thermal thors found instead that the amount of cold gas depends on how instability, even at surprisingly low magnetic field strengths. This tcool/tff varies with radius. greatly broadens the radial range in which halo gas can be thermally (iii) Magnetic field topology. We have seen that straight horizon- unstable. It is a potentially promising explanation for the ubiquity tal and vertical magnetic fields have comparable effects in enhanc- of cold gas in absorption line spectra of the CGM. ing thermal instability. This suggests that magnetic field orientation Our main results are as follows: does not play a significant role and that a tangled field would give cool similar results, although the case λB <λA (where λB is the mag- (i) Magnetic fields have a strong influence in enhancing the am- netic coherence length) deserves more careful study. It may well be plitude of density fluctuations due to thermal instability, and its abil- cool that the characteristic length scale should be min(λA,λA ,λB). ity to fragment into multiphase medium. This enhancement appears (iv) Hydrostatic and thermal equilibrium. Our model is by con- even at very weak magnetic fields, and the MHD results match the − struction always in global hydrostatic and thermal equilibrium. hydrodynamic results only at β ∼ 1000. Otherwise, δρ/ρ ∝ β 1/2, While this is clearly an excellent approximation for galaxy groups where β ≡ Pgas/PB [see equation (9)]. This enhancement arises and clusters, it is potentially more questionable at galaxy scales, par- because MHD stresses weaken the effective gravitational field, and ticularly for lower mass ‘cold mode’ haloes which may lack a hot thus reduce the magnitude of buoyant oscillations (which otherwise hydrostatic halo, or in systems where galactic winds drive a strong disrupt condensing cold gas) by supporting overdense gas against outflow. While local thermal instability is suppressed in a cooling gravity. This can be seen in both particle tracer plots and slice plots flow, even weak magnetic fields enable thermal instability to once of linearized perturbed forces. again operate (Loewenstein 1990;Balbus1991; Hattori, Yoshida & (ii) Magnetic fields have a strong influence on the morphology of Habe 1995). It would be interesting to study if magnetic fields cold gas filaments. The strength and orientation of magnetic fields can play a similar role in modifying thermal instability in galactic strongly influence the size and orientation of cold filaments, even winds. Thermal instability in galactic winds has been the subject of for very weak fields, which can be understood in terms of the MHD much recent interest as a means of explaining cold gas clouds out- stresses. In particular, the spacing between filaments scales with the cool = flowing at high velocity; since they are born at high velocity, they characteristic scale lA vAtcool (see below). can evade the destruction by Kelvin–Helmholtz instabilities which (iii) Thermal instability in the presence of magnetic fields im- cool = would take place if they were accelerated by the wind (Martin et al. prints a characteristic scale lA vAtcool on cooling gas. This scale 2015; Thompson et al. 2016; Scannapieco 2017). is evaluated for conditions in the hot medium (unlike the shattering (v) Large density perturbations. Our simulations start with tiny length cstcool, which is evaluated for conditions in the cool medium; density perturbations, from which thermal instability develops and McCourtetal.2018), since it determines the characteristic scale is independent of a sufficiently small magnitude of initial pertur- in the hot medium at which buoyant oscillations are suppressed bations. However, the density perturbations in galaxy haloes might and gas cooling is enhanced. The appearance of this characteristic be potentially large due to strong turbulence or galactic winds. scale can explain a number of features in our simulations: the scal- − Pizzolato & Soker (2005) and Singh & Sharma (2015) suggest that ing of density fluctuations δρ/ρ ∝ β 1/2 [by setting the scale on 2 cool ∼ thermal instability can develop even at large tcool/tff when initial which magnetic tension operates; B /λA δρg equation (19)], the scaling of filament sizes with β and tcool/tff in simulations, and the ‘overcooling’ effect seen when the box size is smaller than this scale, implying that all modes in the box are destabilized by 9 Previous such simulations have found little difference in simulations with magnetic fields. and without anisotropic conduction (McCourt et al. 2012). Note that this (iv) To zeroth order, the amplitude of density fluctuations and simulations were initialized with extremely weak fields (β ∼ 106), so they did cold gas fraction are independent of magnetic field orientation and not see the dynamical effects we report here. In addition, Wagh, Sharma & McCourt (2014) employed anisotropic thermal conduction and tested differ- cooling curve shape. The independence from field orientation arises ent field geometry including tangled fields, finding little difference between because magnetic tension can be shown to either support overdense hydro and MHD cases as well. This is most likely because the field strength gas directly (for initially horizontal fields) or indirectly (by confin- 3 is also weak (β ∼ 10 ), and vAtcool is much smaller than the length scales ing pressurized hot gas, which in term supports the overdense gas discussed here. or tilts the field lines so that magnetic tension support is important).

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