It's Cloud's Illusions I Recall: Mixing Drives the Acceleration of Clouds from Ram Pressure Stripped Galaxies
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It’s Cloud’s Illusions I Recall:Mixing Drives the Acceleration of Clouds from Ram Pressure Stripped Galaxies S tephanie T onnesen and G reg L. B ryan
Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA Department of Astronomy, Columbia University, 550 W 120th Street, New York, NY 10027, USA
ABSTRACTRam Pressure Stripping can remove gas from satellite galaxies in clusters via a direct interaction betweenthe intracluster medium (ICM) and the interstellar medium. This interaction is generally thought of as a contactforce per area, however we point out that these gases must interact in a hydrodynamic fashion, and argue that thiswill lead to mixing of the galactic gas with the ICM wind. We develop an analytic framework for how mixingis related to the acceleration of stripped gas from a satellite galaxy. We then test this model using three "wind-tunnel" simulations of Milky Way-like galaxies interacting with a moving ICM, and find excellent ageementwith predictions using the analytic framework. Focusing on the dense clumps in the stripped tails, we find thatthey are nearly uniformly mixed with the ICM, indicating that all gas in the tail mixes with the surroundings,and dense clumps are not separate entities to be modeled di ff erently than di ff use gas. We find that while mixingdrives acceleration of stripped gas, the density and velocity of the surrounding wind will determine whether themixing results in the heating of stripped gas into the ICM, or the cooling of the ICM into dense clouds. Keywords:
Ram Pressure Stripped Tails (2126), Intracluster Medium (858), Galaxy evolution (594), Hydrody-namical Simulations (767) INTRODUCTIONAs satellite galaxies orbit within a cluster, their interstel-lar medium (ISM) may interact directly with the intraclustermedium (ICM), the hot halo of gas bound by the cluster grav-itational potential. One of the most commonly invoked ISM-ICM interactions is ram pressure stripping, initially outlinedby Gunn & Gott (1972; GG72). This work introduced thestraightforward competition between a contact force per unitarea exerted by the ICM ( P ram = ρ v ) competing with a grav-itational restoring force per area from the galaxy (2 π G Σ ∗ Σ g ),which remains a commonly used measure today for deter-mining whether a galaxy is likely to be ram pressure strippedand to what radius (e.g. Ja ff e et al. 2018; 2019). Indeed, sim-ulations have verified that using this equation to determinethe stripping radius (the radius outside of which gas will beremoved) is reasonably accurate (Roediger & Bruggen 2007;Tonnesen & Bryan 2009).Thinking of ram pressure as physically pushing gas out ofa galaxy is extremely useful, and can explain several obser-vations of ram pressure stripped galaxies. For example, as Corresponding author: Stephanie Tonnesenstonnesen@flatironinstitute.org predicted by the GG72 picture, gas has been observed to bestripped from the outside-in (Gullieuszik et al. 2017; Pap-palardo et al. 2010; Abramson et al. 2011; Merluzzi etal. 2016; Fossati et al. 2018; Cramer et al. 2019). Inaddition, high-density gas with a higher Σ g may survive inthe disk while lower-density gas is stripped. This has beenobserved in molecular clouds surviving in the disk of NGC4402 (Crowl et al. 2005), and predicted in simulations (Ton-nesen & Bryan 2009). This di ff erential stripping may be fol-lowed by di ff erential acceleration, in which low density gasis pushed to higher velocities than more dense gas (Tonnesen& Bryan 2010; Jachym et al. 2017).However, thinking of the boundary between the ICM andISM as impermeable denies the nature of the hydrodynamicinteraction between the two fluids. This boundary is oftenunstable to a variety of hydrodynamic or hydromagnetic in-stabilities, which can drive mixing and the development of in-termediate temperature and density gas at constant pressure(Begelman & Fabian 1990). For example, Rayleigh-Taylorand Kelvin-Helmholtz instabilities may completely destroya cold clump subjected to a hot wind (Chandrasekhar 1961;Agertz et al. 2007). In addition, non-ideal processes maydrive mixing: heat conduction can evaporate cold gas intothe ICM (Cowie & Songaila 1977; Cowie & McKee 1977).Shock heating of the cool gas may allow these mixing pro- a r X i v : . [ a s t r o - ph . GA ] F e b cesses to act more quickly in stripped tails. In addition toheating cold gas, radiative cooling can lead to entrainment ofthe hot ICM onto cold gas and mixing in surviving cold gasclouds (Klein et al. 1994; Mellema et al. 2002; Scannapieco& Brüggen 2015; Gronke & Oh 2018, 2019).In addition, if we apply the ram pressure model simplyrequiring ram pressure to overcome the gravitational restor-ing force per area on small scales, then it becomes clear thatdense clouds, with their large restoring forces and low sur-face areas, should not be susceptible to stripping, leavingdense molecular clouds impervious to ram pressure. Nev-ertheless, both observations (Moretti et al. 2018, 2020;Sivanandam et al. 2010; Jachym et al. 2014, 2019; Crameret al. 2019) and simulations (e.g., Tonnesen & Bryan 2010)demonstrate that dense gas is present in the stripped tail. This puzzle has been remarked on in other contexts – for ex-ample, Thompson et al. (2016) pointed out (in the contextof cold clumps in galactic winds) that clouds are destroyedmore rapidly than they are accelerated to the surroundingwind speed, at least in the absence of strong radiative cool-ing, a statement which has been verified by high-resolutionsimulations (Schneider & Robertson 2018).There is also observational evidence of gas mixing instripped tails. When comparing the expected gas mass of rampressure stripped galaxies to the mass of gas that is observedin the disk and tail, observers often find lower masses (e.g.Vollmer & Huchtmeier 2007; Ramatsoku et al. 2019; 2020),possibly indicating stripped gas is mixing into the ICM. Ob-servations also find that the metallicity in the stripped gas isoften between that of the ISM and ICM (Fossati et al. 2016;Gullieuszik et al. 2017; Bellhouse et al. 2019).In this paper, we propose a model of ram-pressure that isdriven by mixing processes, rather than a traditional contactforce exerted between by the low-density wind and the high-density cloud. In particular, we write down a few straightfor-ward relations based on this supposition (and some simpleideas about energy conservation) which describe how galac-tic gas is removed and accelerated from a disk in a ram-pressure-stripping wind. We then compare it to the accel-eration of gas from galaxies in wind-tunnel ram-pressure-stripping simulations. In our comparison we examine all ofthe stripped gas, but also focus in one section on dense cloudsthat therefore have longer conduction, viscous stripping, andKelvin-Helmholtz timescales. Thus we are focusing on thegas that is the most likely to remain unmixed with the ICMin a classic “pushing" scenario. The ram-pressure literature sometimes di ff erentiates between ram-pressureand viscous-stripping, with the implication that a non-ideal viscous forceis responsible for removing denser material; however, a coherent physicalpicture of viscous stripping has not been developed to date. This idea, that the acceleration of cold clouds by hot flowsis fundamentally driven by mixing, has received some addi-tional support by two recent works, one examining the in-flow of cold gas on to a galactic disk (Melso, Bryan & Li2019), and another exploring galactic outflows at high reso-lution (Schneider et al. 2020).We begin by introducing our mixing model for gas strip-ping and acceleration in Section 2. In order to test the predic-tions from our model, in Section 3 we describe our simula-tions (Section 3.1) and how we identify and measure proper-ties of clouds in the tail (Section 3.2). We then test our modelpredictions through comparisons with all the gas behind thesimulated galaxies as well as with denser clouds (Section 4).In Section 5 we discuss the survival of the clouds in our sim-ulated stripped tails. We use cloud properties to determine towhat extent the surrounding ICM influences stripped gas inSection 6. We discuss the implications of our results in Sec-tion 7, focusing on caveats in Section 7.4. Finally, in Section8 we summarize our conclusions. GAS ACCELERATION VIA MIXINGIn this section we introduce a simple mixing model to tryto describe the properties of a cloud of mass δ M in the down-stream wake. The cloud is assumed to be a coherent identitythat is identified as a single (cold) object in the wake (to in-form comparison with the simulations below). This cloudcan be thought of being composed of gas with two sources:(1) the cold ISM, at rest with respect to the galaxy (our ref-erence frame), and (2) the hot ICM wind with velocity v wind .We denote the masses of those two sources contributing tothe cloud as M ISM and M ICM , respectively. Note that the gasfrom these two sources may not come from contiguous re-gions in either the ISM or the ICM. Mass conservation im-plies M ISM + M ICM = M total . Defining fraction mass contri-butions f ISM = M ISM / M total and f ICM = M ICM / M total , we canrewrite this as f ISM + f ICM = . (1)Determining the resulting velocity of the gas which endsup in the cloud is, of course, more challenging (which we as-sume to be well mixed, a question directly addressed in thesimulation investigations explored later in the paper). How-ever, our overriding assumption is that acceleration of thecloud is done through the process of mixing, and that mix-ing of the hot (wind) phase into the cold cloud leads to a gainof mass, momentum and energy.Here, we explore two simple models, one based on mo-mentum conservation neglecting the potential of the hostgalaxy, and a second approach based on energy conservation.If we ignore any external forces on the gas during the mix-ing process (as well as any gravitational torques), then a sim-ple application of momentum conservation implies that thecloud velocity would be given by v m = f ICM v wind (2)where we use subscript m to denote the momentum conser-vation estimate (see also Gronke & Oh 2018; Schneider etal. 2020). In principle, we could include the gravitationalforces, but this would require a knowledge of the trajectoryof the gas elements throughout it’s evolution, which requiresmore assumptions. In addition, v wind should more accuratelybe the velocity after being processed through the bow shock(if present), which reduces the momentum of the gas by a fac-tor which depends on the Mach number M = v wind / c s (where c s is the sound speed in the hot gas). For low Mach numbersand subsonic flow this is negligible, and even for the highestvelocity case we explore, we find that the ICM flow veloc-ity in the vicinity of active cloud entrainment is only mildlyreduced from v wind .The other extreme is to assume a form of energy conser-vation. On first blush, this seems problematic, both becauseradiative loses are clearly important and also because of workdone on this gas during its evolution, however we can makesimple assumptions as to the amount of energy lost due to ra-diative cooling as well as assume that no work is done duringthe mixing. We express this with: f ICM v (cid:32) + γ − M (cid:33) − f ISM v = v e + f ICM χ (cid:32) γ − M (cid:33) v (3)The left-hand side is the ‘before’ state; the first term repre-sents the total enthalpy plus kinetic energy of the wind gas,while the gas in the galaxy only contributes a term due toits gravitational potential energy, expressed in terms of theescape velocity ( v esc ). We neglect any gravitational contri-bution for the ICM wind as well as for the cloud after strip-ping, under the assumption that they are su ffi ciently far fromthe galaxy. In the ‘after’ state on the right-hand-side, we as-sume that the gas is cold and so only has a kinetic compo-nent (in this case, we denote the final cloud velocity as v e toemphasize the energy formulation behind the estimate). Weaccount for the radiative losses with the second expressionon the right-hand-side. Our ignorance as to the amount ofenergy radiated is expressed in terms of the ratio to the in-coming enthalpy, such that χ = χ = v e = (cid:16) f ICM v − f ISM v (cid:17) / = (cid:16) f ICM (cid:16) v + v (cid:17) − v (cid:17) / (4) This implies a minimum amount of mixing from the ICMwind to launch a cloud, since v e only becomes real and pos-itive when f ICM > f ICM , crit = v / ( v + v ). For f ICM values below this critical fraction, we assume v e = f ICM (or low v wind ) when we can’t ignorethe gravitational deceleration, but to be increasingly accu-rate for unbound gas. On the other hand, when the energyformulation exceeds the momentum prediction, that impliesthat thermal pressure gradients are accelerating the gas anddoing work above and beyond the momentum content of theinflowing gas. This seems unlikely when mixing is operat-ing in the strong cooling limit, as high-resolution models ofthe hot / cold interface show no pressure gradient (Fielding etal. 2020). Therefore, we expect the energy conservation ar-gument to fail for high values of f ICM . One simple way tocombine this to estimates is simply to take the minimum pre-dicted velocity of each: v cloud = min ( v m , v e ) (5)It is this simple model that we will compare to simulations inthe rest of this paper.The cloud velocity in this model depends on the ICM frac-tion of the clouds ( f ICM ), in addition to the wind and escapevelocities. We illustrate these relationships in Figure 1. Theprimary (thick black) curve in this plot shows the model justderived (Eq. 5), with dashed and dot-dashed thin lines show-ing v m and v e , respectively. We also explore changing v wind and v esc : a perusal of the three lines in this illustration showsthat the velocity-ICM fraction relationship is a ff ected in dif-ferent ways by these parameters. We vary the escape velocityby changing the cylindrical radius from which gas is stripped(called the stripping radius throughout the paper) –in the fidu-cial model (black) this is 20 kpc, and we reduce this to 2.5kpc in the silver comparison line in the cartoon (a perusal ofthe galaxy potential described in Section 3.1 and Tonnesen& Bryan (2009) connects the disk radius to the escape veloc-ity). The wind velocity increases from 1000 km / s in the fidu-cial case to 1500 km / s for the “fast wind" comparison (greyline). We see that changing the radius from which gas is be-ing stripped more strongly a ff ects the velocity-ICM fractionrelationship at low velocities and levels of mixing, as the sil-ver line lies along the black line (seen as the narrower line).However, changing the wind velocity increases the velocityof stripped gas at all mixed fractions.In its most simple form, as described above, our modeldoes not predict the distance of the cloud from the disk,largely because this depends of the rate of mixing of the gasin the past. However, if we assume a mixing rate for cloudswe can make a straightforward prediction for the height ofthe cloud from the disk. We make the simplest possible as- Figure 1.
An illustration of our analytic model, showing the pre-dicted gas velocity versus the fraction of gas originating in the ICM(a gas cloud at f ICM = f ICM = cloud ( f ICM ) from Equation 5, while the dashed and dash-dot linesshow where v m and v e are larger than v cloud , for the fiducial model(see text). In addition to the fiducial model, we show a model witha fast wind (grey line) and one in which the gas is stripped froma smaller radius (silver line), where it must escape from a locationdeeper in the potential well. We also use colored segments to showthe locations along these tracks corresponding to the f ICM value thatgas at a height ranging from 12 to 18 kpc (above the disk) wouldhave for di ff erent assumptions of the cloud mass and wind velocity,as noted in the legend. sumption, that clouds mix at a constant linear rate: f ICM ( t ) = (cid:32) tt final (cid:33) f ICM ( t final ) (6)Here t is the current time and t final is the time at which thecloud reaches f ICM ( t final ). Mixing occurs by the accretion ofall of the wind that hits the cloud. The time it takes to reach achosen f ICM ( t final ) depends on the cloud mass ( M cloud , final ) and(e ff ective) cross-sectional area ( A cloud ), as well as the windvelocity ( v wind ): t final = f ICM ( t final ) M cloud , final ρ wind A cloud v wind (7)We note that our assumptions that all ICM mass that hitsthe cloud is accreted and that A cloud is constant with time aresimplifications. As discussed in Fielding et al. (2020) shearis likely important for mass transfer, as well as the cloudshape (Ji et al. 2019; Gronke & Oh 2020). Then, because we have the velocity as a function of themixed fraction, we have the velocity of the cloud material as a function of time. v cloud ( t ) = v wind f ICM tt final t > t (cid:104) f ICM tt final ( v + v ) − v (cid:105) / t > t > t t > t (8)where t = t final v / ( f ICM ( v + v )) is the time required for f ICM ( t ) to reach the critical value mentioned earlier ( f ICM , crit ),and t = t final v / ( f ICM v ) is the time when the momentumand energy predictions are the same (i.e. v e ( t ) = v m ( t )).Finally, we can solve for the distance ( z cloud ( t )) by integrat-ing the velocity over time: z cloud ( t ) =
23 ( v f ICM q ) / t final (cid:32) t t final − qf ICM (cid:33) / + v wind f ICM t final (cid:16) t − t (cid:17) (9)where q = v / ( v + v ). If t < t , then the last term isdropped and t → t .These relationships are illustrated in Figure 2, where weshow the impact on the position of a fiducial cloud as wevary one property of the cloud at a time. Unlike Figure 1,this is a single snapshot, so the height of each cloud variesdepending on its properties. When an ICM wind has beenacting on a galaxy, gas at a smaller radius is deeper in thegalaxy potential and therefore has higher v esc (as discussedin Section 3.1 and Tonnesen & Bryan 2009), so in the sameamount of time (and for fixed f ICM ) it will not make it as faras our fiducial cloud. In the paper we call this the strippingradius, and in Figure 2 this is shown as R strip . Also, a moremassive cloud will be closer to the disk (red in Figure 2),while a larger radius cloud (with fixed mass) will be fartherfrom the disk (yellow in Figure 2).Using these simple assumptions, we can return to Figure 1and determine where along the ICM-fraction - velocity trackclouds will be as a function of height, using equation 9. Inour schematic diagram we illustrate this relation using theheight range from 15 kpc. We vary the three parameters aslisted in Figure 2 so as to show the impact of v esc , cloudradius, and cloud mass, modifying one parameter at a timewhile keeping the others fixed. The orange bars describe 10 M (cid:12) clouds with 80 pc radii being accelerated by the fidu-cial density wind. Less massive clouds mix and acceleratemore quickly than heavier clouds (compare the orange to red We stress that we are not trying to model the cloud as a single entity duringits evolution, since the gas which ends up in a cloud at a given t final mayhave a complicated previous history and is likely to come from multiplepaths; therefore, these quantities should be thought as “e ff ective" values,characteristic of the cloud’s gas history. fiducial cloud: r cloud = 80 pc M cloud = 10 M sun R strip = 20 kpc large v esc cloud: r cloud = 80 pc M cloud = 10 M sun R strip = 2.5 kpc
10x massive cloud: r cloud = 80 pc M cloud = 10 M sun R strip = 20 kpc r cloud = 200 pc M cloud = 10 M sun R strip = 20 kpc Edge-on slice of galaxy during stripping à high z(t) à low z(t) R strip, small R strip, fiducial à low z(t) Figure 2.
An illustrative cartoon showing the importance of cloud properties for the acceleration of gas from a galaxy due to ram pressurestripping, as predicted by Eq. 9. For context, the background greyscale is the maximum density within a 2.4 kpc slab from run HDHV, taken ata single snapshot in time. Cartoon “clouds" (shown as colored circles) with di ff erent properties are depicted on top of this map and are labelledby their properties. Each cloud di ff ers from the fiducial cloud (orange) in one parameter, with the others held constant: in particular, we fix f ICM , and vary v esc (orange and silver), mass (red), and radius (yellow); we have written how each variable a ff ects the distance as a functionof time ( z ( t ) from Equation 9) for fixed f ICM . The colors are chosen to match those in Figure 1 (we use purple to di ff erentiate the large v esc parameters, although the cloud is orange and silver as in Figure 1). Note that we are using this image only as a background to show the context;the individual colored circles do not refer to specific clouds in this simulation. regions along the black fiducial relation), and clouds withlarger radii mix more quickly than smaller ones (compare theorange to yellow regions).In addition, in Figure 1 we change two wind parameters:the wind velocity and density. We have already discussed thatchanging the wind velocity changes the whole velocity-ICMfraction track, and we also see that this results in gas mov-ing more quickly at any height above the disk (compare thetwo orange regions in the black and grey lines). Also, with afaster wind, less mixed-in ICM mass is required to acceleratethe gas to the same height, so the ICM fraction of a cloud isreduced with respect to a cloud. Increasing the density of thewind increases the rate of mixing and energy / momentum in-put into the cloud, so given the same wind velocity a denserwind will result in clouds moving more quickly and beingmore well-mixed at any height above the disk (compare theorange and cyan regions along the black fiducial cloud rela-tion).Having developed and explored this simple model, in thenext section, we carry out a set of high-resolution simulations of ram-pressure stripping with three di ff erent wind parame-ters and explore how well it performs. METHODTo follow the gas, we employ the adaptive mesh refine-ment (AMR) code Enzo (Bryan et al. 2014) which solves thefluid equations including gravity and optically thin radiativecooling. The code begins with a fixed set of static grids andautomatically adds refined grids as required in order to re-solve important features in the flow. Our simulated region is300 kpc on a side with a root grid resolution of 256 cells. Weallow an additional 5 levels of refinement, for a smallest cellsize of 37 pc. The refinement criteria is based on gas mass,with a resolution of ∼ . × M (cid:12) (HDHV), ∼ . × M (cid:12) (HDLV), and ∼ . × M (cid:12) (LDLV), meaning thatwhenever a cell exceeds this mass it is refined into 8 smallsub-cells. The simulation includes radiative cooling usingthe GRACKLE (Version 3) cooling tables including metalcooling and the UV background from HM2012 (Smith et al.2017). 3.1. Simulation Initialization
Our galaxy is placed at a position corresponding to(150,150,75) kpc from the corner of our cubical 300 kpccomputational volume, so that we can follow the strippedgas for more than 200 kpc. The galaxy remains stationarythroughout the runs, with the disk aligned in the x-y plane.The ICM wind flows along the z-axis in the positive direc-tion, with the lower x, y, and z boundaries set for inflow andupper x,y, and z boundaries set as outflow.We model a massive spiral galaxy with a flat rotation curveof 200 km s − . It includes a gas disc that is resolved tothe maximum level (37 pc). The galaxy model also in-cludes the static potentials of the stellar disc, stellar bulgeand dark matter halo, directly following the set-up of Roedi-ger & Bruggen (2006). Specifically, we model the stellardisc using a Plummer-Kuzmin disc (see Miyamoto & Nagai1975), using a radial scale length of 3.5 kpc, a vertical scalelength of 0.7 kpc and a total mass of 1 . × M (cid:12) . Thestellar bulge is modeled using a spherical Hernquist profile(Hernquist 1993) with a scale length of 0.6 kpc and a totalmass of 10 M (cid:12) . The dark matter halo is modeled using thespherical model of Burkert (1995), with an equation for theanalytic potential as given in Mori & Burkert (2000). Thedark matter halo has a scale radius of 23 kpc and a centraldensity of 3 . × − g cm − . We describe our disk setup indetail in Tonnesen & Bryan (2009, 2010).To identify gas that has been stripped from the galaxy wealso follow a metallicity value that is initially set to 1.0 insidethe galaxy and 0.3 outside. Because we do not include starformation in this simulation, the metallicity can also be usedto determine the origin of the gas in each cell, in particular wecan track the ratio of galactic gas to ICM gas on a cell-by-cellbasis.In this paper we discuss three simulations. In all three runs,as in our earlier work (Tonnesen & Bryan 2009, 2010, 2012),we impose a delay of 100 Myr before the wind enters thebox in order to allow multiphase gas to self-consistently de-velop in the disk via radiative cooling. All of the ICM windshave temperatures of 7.08 × K. We vary the velocity anddensity as shown in Table 3.1. The low density, low velocitywind (LDLV) has a velocity of 1000 km s − ( M ∼ × − g cm − . The high density, low velocitywind (HDLV) has a velocity of 1000 km s − ( M ∼ × − g cm − . Finally, the high density,high velocity wind (HDHV) has a velocity of 3230 km s − ( M ∼ × − g cm − . The mini-mum and maximum ram pressure parameters were chosen toroughly correspond to the ICM wind parameters of two jel-lyfish galaxies observed in the GASP sample (GAs StrippingPhenomena in Galaxies with MUSE; Poggianti et al. 2016):JO204 (LDLV) and JO201 (HDHV) (Gullieuszik et al. 2017;Bellhouse et al. 2017), with the middle simulation a test of Name Velocity Densitykm s − g cm − LDLV 1000 5 × − HDLV 1000 1.2 × − HDHV 3230 1.2 × − Table 1.
The wind velocity and density of the three simulationsdiscussed in this paper.
Figure 3.
The distribution of the minimum temperature found inclumps in the all three simulations (number of clouds per logarith-mic temperature bin). The colors denote height above the galaxydisk. We have not di ff erentiated between the three simulations asthey all have a similar minimum in their temperature distribution.The y-axis scale is set to log to highlight the minimum in the dis-tribution at all heights above the disk, at about 30 000 K (dashedvertical line). Clouds are required to have a minimum temperaturebelow this value. the impact of changing a single wind variable rather than bothdensity and velocity.3.2. Cloud Selection
In this paper, we examine both the state of all of the gas inthe wake (Section 4.1) as well as focusing just on the densegas (Section 4.2), which is more observationally accessible.In order to find higher-density “clouds" in our tails, weused the clump finder routine in yt (Turk et al. 2011). Thisuses level sets to find connected cells with values above agiven selection criteria. We searched for clouds using the gasdensity, starting with 10 − g cm − and increasing the den-sity by a factor of two for each level set. This minimum den-sity is relatively arbitrary, and we use other characteristics ofthe clumps, as described below, to make a more physically-motivated cloud selection. In brief, we required a minimumnumber of cells and a low minimum temperature in the clumpto include it in our analysis.In more detail, the clump finder algorithm will find clumpsdown to a single cell, and indeed, we find that about half ofour raw clump sample have fewer than 10 cells. In order toeliminate clumps that are actually small density perturbationsin the di ff use stripped tail, we only keep those identified withat least 10 cells in total (we note that we have repeated thiswork using clumps with at least 300 total cells and find thesame trends, with some results shown in Appendix A).We also find that clumps have a bimodal distribution intheir minimum temperature, as shown in Figure 3. As we areattempting to choose cold, dense clouds rather than simpleoverdensities in the tail gas, we only include clumps whoseminimum temperature is below 30 000 K.We note that, when we impose these two selection criteria,the lowest maximum density of any cloud across all threesimulations is more than 4 × − g cm − , so our minimumsearch density (10 − g cm − ) does not have a strong impacton the number of clouds we identify. To orient the reader, adensity of 4 × − g cm − will be refined to ∼
300 pc, andonly densities reaching ∼ × − g cm − will be refined to37 pc, i.e. the centers of the most dense clouds.During the clump finding procedure we also save severalcloud properties. We find the maximum and minimum val-ues of density, temperature, and the ICM fraction within thecloud. Using all of the cells identified as belonging to ourclouds, we also save the mass-weighted mean cloud positionand velocity in addition to the physical characteristics listedabove.In summary, our final set of “clouds" are ten or more con-nected cells consisting of gas at least an order of magnitudedenser than the ICM that have minimum temperatures morethan three orders of magnitude lower than that of the ICM.This makes us confident that our clouds are physically dis-tinct entities rather than just being ephemeral overdensitiesin the wake.We identify these clouds in three evenly spaced regions:12-18 kpc above the disk, 82-88 kpc above the disk, and 152-158 kpc above the disk. Throughout the paper these heightswill be identified as 15 kpc, 85 kpc, and 155 kpc. These re-gions were selected so that we can easily compare gas prop-erties as a function of height above the disk, and because theyspan the range of tail lengths seen in observations. We testedour results using di ff erent height bins (10-20 kpc, 76-86 kpc,and 160-170 kpc) with no qualitative change to our results.We perform this clump identification at each output (10Myr apart) for each simulation. We do not attempt to followany individual clump, but identify the population of clumpsin our simulations in the three regions at each output.In Figure 4 we show density projections of “early" and“late" illustrative outputs from each simulation, with coloredlines indicating the three narrow height ranges we analyze,as described above. Here we are using “early" and “late" todenote the timeline of the development of the tail. Specifi-cally, the “early" output is close to the output at which the most clouds are found in the 15 kpc region, and the “late"output is close to the output at which the most clouds arefound in the 155 kpc region (as seen in Figure 10). In orderto directly compare the simulations, and highlight di ff erencesin the tails, we chose to show the 360 Myr snapshot for allthree. This same output is also used in Figures 5 - 7.The tail structure varies as a function of height and timewithin a simulation, and di ff ers across simulations at thesame time. At earlier times clouds are tightly packed in thetails, while at later times clouds are less densely distributed.This is true even looking at a single height above the disk (forexample within the red 15 kpc region). The wind propertieshave a significant impact: for example, at 360 Myr, dense gasin the HDHV tail has reached 155 kpc from the disk, whilein the LV runs it has barely reached 85 kpc. Even within theLV runs the gas distribution is di ff erent: denser clumps areseen at larger distances in HDLV than in LDLV. We also notethat the ICM of the HD runs is denser than in LDLV, leadingto the darker background density seen in the projections. TESTING THE MIXING MODELOur simple analytic mixing model, developed in Section 2,predicts a relationship between stripped cloud velocity andmixed fraction, modulo wind velocity and galaxy escape ve-locity. In this section we use the three simulations describedin Section 3.1 to test these predictions.4.1.
All Gas
We begin by looking at the properties of all gas in the tailbefore turning to dense clouds (which connect more directlywith observations) in the next section. The gas velocity inour mixing model depends only on the ICM fraction of gas(for fixed v esc and v wind ), and therefore can be tested using allof the gas behind the galaxy in our simulations. According toour model, this “tail" gas will consist of a range of ICM frac-tions, from nearly pure galactic gas that was loosely boundto the galaxy and moving slowly, all the way to nearly pureICM gas moving at the wind speed.In Figure 5, we see that, indeed, gas behind the galaxy hasa range of ICM fractions. In this figure we plot the velocity-ICM fraction relation in the HDHV run at three heights abovethe disk (15 kpc, 85 kpc, and 155 kpc). Each of these plotsuse an output taken shortly after any dense clouds are iden-tified at those heights. In black we overplot the analytic re-lation between ICM fraction and velocity using a strippingradius of 3 kpc (used to determine v esc ), as this is about theradius of the gas disk at the end of the simulation. We seethat at any height above the disk, gas tends to have a broadrange of ICM fractions, and generally falls along the ana-lytic relationship between ICM fraction and velocity. As wemove farther from the disk, the minimum velocity and ICMfractions are shifted to higher values. This agrees well with Figure 4.
Images of projected density for an early (upper panels) and late (lower panels) output from each of our simulations. Each projectionis 185 kpc by 110 kpc. The regions on which we focus in this paper are denoted by the colored lines (which are reflected in later figures). Gasbehind galaxies shows a range of densities and tail morphologies as a function of height above the disk, time since stripping began, and windparamaters.
Figure 5.
The gas velocity- f ICM relation at three di ff erent heights(and times), as labelled, above the disk in run HDHV. There is moremixing of gas as we move farther from the disk. Despite this, thegas at all heights falls along a similar velocity - ICM fraction line.The solid line in each panel is from equation 5. Figure 6.
All gas between 15 kpc in HDHV at an early time (170Myr) and later time (360 Myr). The black line shows the modelprediction using a stripping radius of 3 kpc. At early times there isgas that is less mixed than at later times. More energy is requiredto remove gas from inner radii, which results in the higher ICMfractions at later times. our model prediction that continual mixing occurs as gas isdriven away from the disk, therefore our lowest ICM frac-tions will be found near the disk.We note that the tail gas from 15 kpc in HDHV shows amaximum velocity ( ∼ / s) that is lower than our in-put wind velocity (3230 km / s). This is because the wind ve-locity in HDHV is supersonic and creates a large bow shock,deflecting the incoming wind along the x- and y-axes in addi-tion to the z-axis. At later times, much of the gas is stripped,the bow-shock shrinks, and gas flows more directly along thez-axis, the direction of gas inflow at the edge of the box. As0 Figure 7.
Gas from 15 kpc at an early (top panels) and later (bottom panels) time in the LV runs: LDLV (230 Myr and 360 Myr) and HDLV(210 Myr and 360 Myr). In each run the majority of gas falls near the analytic line. Gas with low ICM fractions tends to be moving with highervelocities at earlier times. Also, in comparison to the HDHV run, the LV gas tails show a much broader velocity distribution at a given ICMfraction. we show below, the slow velocities in the z-direction of high f ICM gas is not a long-lived e ff ect.In Figure 6 we examine how the velocity-ICM fraction re-lation changes over time in our simulations. We plot thisrelation in the HDHV run 15 kpc above the disk at an early(170 Myr) and late (360 Myr) time. The amount of gas inthe tail changes (as we will explore in more detail later in thepaper), but here we highlight the fact that, at all times in oursimulations, most of the gas in the tail falls approximatelyalong our analytic relation.By examining the minimum ICM fraction of gas from 15kpc as a function of time, Figure 6 illustrates the importanceof the escape velocity in the velocity-ICM fraction relation.We find that at early times some gas is moving slowly with low ICM fractions, while at late times there is only gas mov-ing quickly with higher mixed fractions. This reflects the de-creasing stripping radius, and the resulting increase in escapevelocity, as a function of time. The impact of the strippingradius on the ICM fraction and velocity of gas can be seenin Figure 1 by comparing the orange bars in the fiducial andsmall stripping radius lines.We note that at later times the velocity of low- f ICM gastends to fall below the analytic line. In comparison with thelarger distances in Figure 5, we find that this occurs mostdramatically near the disk. We posit two likely reasons forthis lower velocity gas. First, there continues to be a smallbow shock near the disk, and the gas flow is not completelyin the z-direction. We also find that at early times the disk1is being stripped rapidly and shrinking, but at late times thedisk size stabilizes. Therefore gas that is stripped is morelikely to be near the disk edge and somewhat protected fromthe wind. We discuss this more below.Now that we have carefully compared the HDHV run toour analytic model, in Figure 7 we verify that this relation-ship holds in all three of our runs. Here, in two columnswe have plotted the gas 15 kpc above the disk in LDLVand HDLV, in each case choosing a time shortly after denseclouds are identified (upper panels) and at 360 Myr (lowerpanels) to match the projections in Figure 4. The disk strip-ping radii for the model lines are chosen to be 20 kpc (LDLV)and 15 kpc (HDLV), as these are near the surviving disk ra-dius at early times for the three runs. We note that at earlytimes, gas with a low ICM fraction is moving more quicklythan the analytic line, likely due to gas stripped from a largegalactic radii, while at 360 Myr more of the gas lies alongor below the line, indicating stripping from small radii. Thisagrees well with our model prediction illustrated in Figure 1.The generally good agreement between the model predic-tions and the gas distribution in these plots demonstrate thebasic success of the picture. While the overall predictions areverified, we highlight two points of disagreement betweenthe LV runs and our model predictions. First, at the earlytimes shown in Figure 7, the gas velocity is slightly largerthan predicted, and this is true even for pure ICM gas (at theright edge of the upper two panels), which have speeds inexcess of the inflow velocity (1000 km / s for the LV runs).We see in the lower panels of Figure 7 that this excess veloc-ity is less pronounced at later times. The second, more dra-matic, disagreement between the simulations and the modelin the LV runs is the gas component with high ICM fractionsand low, even negative, velocities (in the lower-right of eachpanel).To look at these discrepancies in more detail, in Figure 8we plot the z-velocity of the gas, as a function of cylindri-cal radius for gas within the same 15 kpc height range ofthe disk in the HDLV run (to match the HDLV panel in Fig-ure 7). Rather than the total mass in each histogram cell, wehave color-coded these two plots using the mean ICM frac-tion (top) and the mean gas density (bottom) in each cell ofthe histogram.First we describe the general gas distribution in these pan-els. In the inner regions, gas only has negative velocities –this indicates the disk’s “shadow", where it blocks the flowof the ICM wind. There is then a region between ∼ Figure 8.
These panels both plot the gas velocity in the wind di-rection against the cylindrical radius (for all gas in the 15 kpc bin)from the disk in the HDLV simulation at 210 Myr into the simula-tion. The upper and lower panels are color coded by the mean ICMfraction and gas density in each histogram cell, respectively. We seethat the high ICM fraction, low density gas in the inner tail region ismoving quickly, and that the low density gas in the disk “shadow"has negative velocities. the disk. Visual inspection suggests that this is due to vor-tices created by the wind flowing past the surviving disk,with some smaller vortices created by dense clouds in thetail. When the vortex velocity aligns with the wind velocitywe see our maximal velocities. At early times a particularlylarge vortex forms that results in a large amount of gas at highvelocity while at later times the majority of ICM-like gas ismoving at the inflow velocity. We do not see these fast flowsin the HDHV run because the galaxy is stripped quickly to a2small radius so there are not large vortices containing a sig-nificant amount of mass.We can also clearly see in Figure 8 that gas with highICM fractions and negative velocities is falling back in theshadow of the disk (i.e. at small galactocentric radius). Itfalls o ff of our predicted f ICM -velocity relation because theprimary mode of acceleration of this gas is gravitational ac-celeration rather than mixing. The resulting infall velocityis of order 300 km / s, a value which is set by the accelera-tion of the galaxy and is unrelated to the wind velocity. Theremaining gas disk is very small in HDHV, and so there isno fallback in that run (as seen by the lack of negative ve-locities in Figure 5). Interestingly, the gas in the shadow ofthe disk is generally low density, which likely allows it to bemore easily shifted to di ff erent radii due to disordered mo-tion or vortices (there may be occasional fallback of cloudsbut this is a minor e ff ect in these runs). When we focus onjust the dense gas in the bottom panel, we see a very clear re-lationship between radius and velocity, again illustrating theimportance of the stripping radius (and thus escape velocity)on gas acceleration.In summary, our simple model is an excellent predictionof the gas velocity-ICM fraction relation, and the di ff erencesbetween the model and simulations can be explained by thehydrodynamical interaction of a wind hitting a disk.4.2. Focusing on Dense Clouds
While it is commonly accepted that low-density gas in rampressure stripped tails mixes with the ICM, dense clouds areoften assumed to consist of galactic gas that maintains its in-tegrity as it is being accelerated by the wind. Therefore, inthis section we focus directly on comparing our model pre-dictions to dense clouds in the tail.As we did with all of the gas in our tail, we compare thecloud velocity to the ICM fraction in our clouds in Figure 9.The contours in each panel show the cloud velocity versusICM fraction for clouds identified at early times in each sim-ulation. The clouds are defined using contiguous regions ofdense gas, as described in Section 3.2, and the contours showthe number of clouds in any bin.We use clouds identified in a limited range of outputs be-cause the radius from which gas is stripped decreases overtime so the e ff ective escape velocity for clouds at a givenheight increases over time. As our model uses a single es-cape velocity, we choose a narrow range in time to decreasethis variation. We also note that we are not tracking individ-ual clouds in our simulations, so we cannot determine the re-lationship between clouds at di ff erent distances and outputs.Therefore, we use the first 4 outputs (5 at 155 kpc in orderto have more than 150 clouds in LDLV) at each height thathave any identified clouds in an attempt to follow a similarpopulation of clouds moving away from the disk. Figure 9.
The velocity as a function of the ICM fraction of clouds.The contours show the density of clouds per linear range in velocityand gas fraction. As in Figure 7, the panels from top to bottom arethe three runs: LDLV, HDLV, and HDHV. Clouds from the first 4(15 and 85 kpc) or 5 (155 kpc) outputs at which they are identifiedin each simulation and height are shown. For the analytic model,the cloud radius is set to 80 pc, and the mass ranges from 2 × to 4 × M (cid:12) . For LDLV and HDLV the wind velocity is set tobe 1000 km / s, and for HDHV the wind velocity is set to be 3230km / s. Simulations are generally in good agreement with the analyticmodel. = y = × to 4 × M (cid:12) for all threesimulations. This is based on the measured mass distributionof our selected clumps, which ranges from about 5 × to1 × M (cid:12) , with the peak of the mass distributions in allsimulations at about 2 × M (cid:12) . A cloud radius cannot bedirectly measured, as the clouds are not spherical. Indeed,we expect them to be extended along the wind direction. Wechoose a radius of 80 pc based on the measured volumes ofthe clouds, recognizing that this could vary by a factor of sev-eral (from 1 cell across to a radius of 600 pc, correspondingto a spherical cloud at our maximum measured volume of 3 × cm ). Using these assumptions allows us to predictwhere along each curve the clouds will fall when 15 kpc, 85kpc, and 155 kpc above the disk (Equations 8 & 9). This isshown by the thick shaded regions along the curves.Despite the simplicity of our model, we again find goodagreement with the simulations. In particular, the f ICM -velocity relations follow quite well the predictions of themodel and there is qualitative agreement in that higher cloudheight corresponds to larger cloud velocities and highermixed fractions. However, there is some disagreement withthe predictions in detail, which we can separate into two gen-eral trends: shifts above or below the model curve, and shiftsalong the model curve.We first focus on the shift of the simulated output o ff themodel curve. First, we note that this only occurs in the twoLV runs near the galaxy disk. As we have discussed with re-gards to Figure 1, there are two main variables that changethe shape of the velocity-ICM fraction curve in our model:the wind velocity and the galaxy escape velocity. Includingthe galaxy escape velocity in our model results in a lower ve-locity at a given ICM fraction than we see in our simulations.However, we note that even simply using a linear relation-ship due to momentum transfer would still predict velocitiesslightly below the peak of the HDLV contours. We next focus on our predictions for where along thevelocity-ICM fraction curve gas should lie as a function ofheight above the disk. We note that the LV runs show goodagreement between the predictions and simulations, althoughthe model has a slight shift towards higher velocities andICM fractions as a function of height. In the HDHV simu-lation, these predictions di ff er much more dramatically, withthe di ff erence between the simulation and model increasingwith height above the disk.This could be for several reasons. First, we assumed aconstant cloud mass, which is likely not the case – cloudscould be constantly accreting gas and fragmenting, processeswhich are not included in our simple model. Indeed, as wediscuss in the next section, we expect that not all clouds sur-vive. On the same note, we assume a constant 80 pc radius.The clouds could be very elongated along the wind direc-tion and be narrower than we model, which would shift ourmodel shaded regions to lower velocities and ICM fractions.Finally, for simplicity we assume that the ICM fraction ofcloud gas changes linearly with time, but as the velocity dif-ference between the cloud and ICM decreases, the mass de-position rate should also decrease, leading to a slower in-crease in the ICM fraction. As we see in HDHV, this shouldlead to a shift in our model towards lower velocities and ICMfractions, particularly at large distances above the disk. CLOUD SURVIVALBeyond our model, the evolution of clouds in stripped tailsis important for understanding gas mixing and cooling aswell as star formation in the ICM. As we have discussed inthe Introduction, there remains debate about whether densegas is removed from the disk and survives in the tail, orwhether dense gas can form within that tail from lower-density galactic gas that is more easily stripped.In this section, we directly address the evolution of cloudsin our simulations, first empirically by examining cloud prop-erties in the simulations as a function of height, and then the-oretically by comparing the cooling and destruction (crush-ing) times of clouds.5.1.
Cloud Number and Mass As a Function of HeightAbove the Disk
Because we cannot track individual clouds in our simula-tions, the most straightforward way to determine if cloudssurvive in the stripped tail is to simply count the number ofclouds at di ff erent heights above the disk. However, becauseclumps could fragment or merge, summing the total massin clouds, or the mass flux, may be a more useful measureof the evolution of dense gas in the tail. All of these met-rics are shown in Figure 10. We note that clouds move at arange of velocities (as shown in Figure 9), and so compar-ing gas at di ff erent heights is not likely to give us the exact4 Figure 10.
The number of clouds (top panels) and the total mass in clouds (bottom panels) as a function of time in the three simulations, ineach of the three height ranges previous defined. Note that the x- and y-axis ranges di ff er across the panels. The bottom panel shows the massflux through each 6 kpc region as a function of time measured from the time of peak flux (Time - T peak ). Although the number and mass ofclouds is highest in each simulation close to the disk, in HDLV the number of and mass in clouds may increase from ∼
85 kpc to ∼
155 kpc, andthe mass flux increases as a function of height above the disk. same cloud population. However, with that caveat in mindwe make rough comparisons using the peak of the distribu-tions at any height.Clearly, in all simulations, the number of clouds decreasesas we compare the 15 kpc height range to either 85 kpc or155 kpc. Thus, many clouds are not surviving intact as theyare being accelerated through the tail by the ICM wind. Inagreement with this interpretation, we also see that the totalmass in clouds decreases as we look farther from the disk. However, when we compare the number of clouds at 85kpc to that at 155 kpc, the fate of clouds becomes less clear.In both the LDLV and HDHV simulations, the number of,and mass in, clouds at their peaks decrease from 85 kpc to155 kpc. In contrast, the peak number of clouds found at 155kpc in the HDLV run is actually larger than the peak found at85 kpc, and this increase in the number of clouds is reflectedin an increase in the total mass in clouds. While clouds maybe both destroyed and formed in any of the simulated tails,the increase in cloud number and mass in the HDLV wind5
Figure 11.
The ratio of the minimum to the maximum ICM fractionversus the minimum ICM fraction for clouds in the three simula-tions. In all runs, the minimum ICM fraction increases as a functionof height above the disk, indicating continued mixing of strippedand surrounding gas. Also, the ICM fraction within each cloudtends to be nearly uniform (and becomes increasingly uniform withheight above the disk). indicates that clouds are likely to be forming in the tail, witha net increase in the number of clouds beyond ∼
85 kpc abovethe disk.Because the cloud velocity varies as a function of heightabove the disk (and there is a range of cloud velocities at anygiven height as shown in Figure 9), we also look at the massflux in clouds for a physically-motivated view of the cloudflow as a function of height. This is shown in the bottompanels of Figure 10; here we have shifted the fluxes for thethree di ff erent heights such that their peaks are coincident at t = ∼
85 kpc to ∼
155 kpc. Even though the total mass ofclouds is increasing, it is not due to stripped clouds simplyaccreting ICM gas. In other words, in all three simulationsthere is no surviving core of pure “galactic material" that isbeing accelerated while the outer layers of the clouds aremixing. We note that this result is robust even when only se-lecting clouds with at least 300 cells, as shown in AppendixA. 5.2.
The Wake as a Whole Figure 12.
Mass versus time for gas in the entire box (“all"), within 10 kpc of the disk plane (“disk"), and from 10-224 kpc above the disk(“wake"). Each panel shows the gas mass for which f ICM < ρ > × − g cm − (green). In each simulation the mixing of disk gaswith the ICM results in a di ff erent amount of dense gas relative to the low- f ICM gas. Note that gas begins to leave the simulation volume after670 Myr for the LV runs and 260 for the HDHV run and so care should be taken interpreting results after those times (shaded to guide the eye).The di ff erence between the green and cyan dashed lines in the HDLV run signifies substantial condensation of ICM gas. As we have shown in the previous section, it is usefulto compare cloud properties at di ff erent distances from thestripped galaxies. However, we must be careful not to as-sume that clouds move together in clumps, or in other wordsthat clouds found in the 15 kpc region at one output will laterbe found together 85 kpc from the disk. This is visually clearwhen we consider the di ff erent distribution of clouds as afunction of time in Figure 4, and is also predicted by the ve-locity range of clouds shown in Figure 9.It is informative, then, to step back and measure theamount of dense and unmixed gas as a function of time inthe simulations as a whole to see if they follow our expec-tations. In Figure 12 we do just that. For each simulation,the mass of gas with f ICM < ρ > × − g cm − is shown in green (i.e. dense gas ofall origins). The linestyles denote di ff erent spatial regions ofthe simulation. The early rapid increase in the dense gas (inthe first 20 Myr of the simulation) comes from the relaxationof the disk, as dense clouds form out of the initially smoothdisk gas.Before exploring the ramifications of these plots, we notea technical point, which is the flow of gas out of the simu-lation volume. Because we are only showing here dense orlow f ICM gas, it is not clear whether decreasing mass corre-sponds to gas leaving the simulation volume, or transitioninginto another state. However, we can resolve this di ff erenceby noting that in both LV runs the total amount of gas in thesimulation decreases after about 670 Myr, and in HDHV thetotal amount of gas in the simulation decreases after about260 Myr, indicating that at these late times more gas is leav-ing the box than is flowing into it (shown as the shaded re-gions in each panel). This explains most or all of the late-timedecrease in wake gas for both HD runs.If we first consider the mass of gas with f ICM < t ∼
200 Myr for the LV runs and t ∼
150 Myrfor the HV run), the total gas mass in the box increases aspure ICM gas is beginning to mix with the disk gas. Com-paring the wake to the disk gas masses, we see that most ofthe increase in gas mass is occurring behind the disk (in thewake). This agrees well with our main prediction that mix-ing drives gas acceleration. We also see a rapid drop in thewake mass as gas that is stripped continues to mix to high f ICM values, as seen in Figure 11.When we examine the dense gas in the simulation we seea somewhat di ff erent story. In all simulations, the amountof dense gas in the disk region decreases as the amount ofdense gas in the wake increases, indicating that dense gas isstripped from the galaxy. However, the total amount of densegas and the ratio between dense and low f ICM gas di ff ers dra-matically in the di ff erent simulations.In comparing the dense gas mass to the low f ICM gas masswe will move from right to left in the panels. In HDHV thereis never an increase in the total dense gas mass after the windhits the disk. When we compare the wake gas mass measuredusing f ICM or by our density cut, the HDHV curves are sim-ilar in width, but the amount of low f ICM gas in the wake ishigher than the amount of dense gas, indicating that in gen-eral mixing is heating stripped gas.However, HDLV seems to tell the opposite story. The to-tal dense gas mass nearly doubles from the initial mass to thepeak mass, and reaches much higher values than the total low f ICM gas. The dense gas peak in the wake is much broaderand higher than the f ICM peak, indicating that as gas mixes itcontinues to cool. This indicates that we are seeing signifi-cant gas condensation out of the ICM, driven by the mixingfrom clouds stripped out of the galaxy.Finally, the LDLV run does not necessarily tell a singlestory of mixing. As with HDHV, the total low f ICM gas riseswhile the total dense gas falls, and in the wake the peak low f ICM gas is higher than the peak dense gas. Interestingly, if7we look closely at the wake gas in LDLV, the mass of densegas has a slightly broader time profile than the mass of low f ICM , and there is more dense gas than low f ICM gas in thewake after 600 Myr. This may speak to late cooling of mixinggas, reminiscent of (but weaker than) the HDLV run.To summarize this section and the previous one, we findthat in all three simulations, f ICM of clouds increases withdistance from the disk, demonstrating mixing. Importantly,this does not seem to correspond to universal cloud destruc-tion or accretion. In more detail, as we move beyond 15 kpcfrom the disk, clouds seem to be destroyed in two simula-tions: the number and mass of dense gas in clouds continuesto decrease throughout the wake and in time in both LDLVand HDHV. However, looking farther in the wake in HDLVwe see that the mass of clouds increases at large distances,and this is reflected in the total dense mass in the wake.5.3.
The Cooling versus Destruction time of Clouds
We also consider the survival of clouds in the stripped tailsmore theoretically by comparing the cooling and destruction,or “crushing" times of the clouds. To calculate the coolingtime of the gas we use the cooling rates from the gracklecooling tables implemented in our simulations. Since, as wehave shown, the metallicity varies little in the clouds, we usethe average metallicity in the cooling calculation.We calculate the cloud crushing timescale as (e.g., Klein etal. 1994): t cc = (cid:114) ρ cloud ρ ICM R cloud v di ff (10)For our calculation, ρ cloud is the maximum cloud density, ρ ICM is the inflowing ICM density, v di ff is the cloud velocityin the wind direction subtracted from the input wind velocity,and we assume spherical clouds and use the cloud volume tocalculate R cloud .In Figure 13 we plot the ratio of the cooling and crushingtimes as a function of mean cloud density. We calculate thecooling times of the clouds with two di ff erent assumptions.In the top panels we use the minimum temperature of thecloud and the maximum density to calculate the maximum Hnumber density using µ =
1. In the bottom panels we usethe “mixing" temperature and density (with µ = .
6) fromGronke & Oh (2018; GO18). The distributions are similar,but we find that low density clouds are closer to t cool = t cc using this second definition, and therefore are more likely tobe destroyed before they cool. This is largely because of thedependence of the cooling rate on density squared, which re-sults in faster cooling rates for higher density gas. Only ourdensest clouds have temperatures cooler than the peak of thecooling curve, so the rate is not generally strongly dependenton temperature. Thus, while the cooling rate is quite similarfor the cloud and the mixing layer, the cooling time is longer in the mixing layer because of the higher temperature. How-ever, using either formalism we (marginally) find that mostclouds will cool before they are destroyed.We also note that the ratio t cool / t cc of clouds at di ff er-ent heights above the disk have very similar distributions.However, in both the HDLV and HDHV simulations, cloudswith higher densities extend to lower t cool / t cc when they areclose to the galaxy. In addition, the fraction of clouds with t cool / t cc > t cool = t cc threshold, and extends to the most extreme (small) t cool / t cc ratios. The HDLV simulation has an ICM wind in whichthe number and mass of clouds increases from ∼
85 kpc to ∼
155 kpc, and in which the dense gas in the entire box in-creases over time. This is qualitatively consistent with thesmall t cool / t cc ratios.How does this theoretical calculation align with our find-ings in Section 5.1, that clouds do not (generally) surviveintact in the tail? In the GO18 picture, even clouds that ac-crete gas first seem to lose dense gas mass, and only then cooland accrete gas in tails very late in the simulation (more thanten cloud crushing times at the highest density ratios). There-fore, it is likely that the clouds that eventually accrete mass ina GO18 scenario are well-mixed throughout, consistent withour Figure 11.However, both the number and mass of clouds in our sim-ulations decreases from 15 kpc above the disk to larger dis-tances. In GO18, higher density ratios between the surround-ings and the cloud result in longer lag times before the densecloud mass starts to increase. In our ICM wind simulation,with higher density ratios than simulated in GO18 (or Gronke& Oh 2019), the growth rate of these clouds may be evenslower (comparing the ICM density to the average cloud den-sities from Figure 13, our clouds range from density ratios of ∼
100 - 10 ). The cloud crushing times range from 3 × years (the shortest crushing time in the HDHV run) to 10 years (the longest crushing time in HDLV). However, the dis-tribution of crushing times in HDHV peaks at 2.5 × yearswhile both LV runs peak between 1-2 × years. Even usingthe maximum cloud velocities from Figure 9, the travel timefrom 15 kpc to 155 kpc above the disk is more than ten cloudcrushing times for the majority of clouds. Therefore, only ifthe growth times of our clouds are more than ten cloud crush-ing times may we need to follow our clouds longer in orderto see growth, or at density ratios of ∼ clouds may notaccrete gas from their surroundings.In all three simulations, as we have shown, our cloudsstraddle the t cool = t cc line. By using the maximum den-sity for our clouds we are choosing the maximum possiblecrushing time for our comparison. Further, we are using themass density of the gas and an estimated µ as a proxy for H8 Figure 13.
The ratio of t cool / t cc versus the mean density of the cloud for all three simulations. As in Figure 9, the contours are based on thedensity of clouds in each two-dimensional histogram bin. In the top panels we use the maximum density and minimum temperature of the cloudto compute the cooling time, and generally find that these clouds should cool. However, in the lower panels we use the mixing temperature anddensity as in Gronke & Oh (2018) and find that these clouds lie much closer to the t cool = t cc boundary (marked as a dashed line). Comparing thenearest (red contours) to the most distant clouds (blue contours), we see that while the distributions of t cool / t cc have a large amount of overlap,in both HD simulations clouds near the disk extend to much lower t cool , mix / t cc ratios. number density in our cooling time calculation. These ap-proximations could shift our clouds above or below the lineof equality. We are performing high resolution simulations ofindividual clouds to follow their evolution in detail and ver-ify these calculations (Smith et al, in prep and Abruzzo et al.,in prep). THE INFLUENCE OF THE ICM ON CLOUDPROPERTIESIn this paper we have used three simulations to support ourmodel for ICM-mixing as the driver for gas acceleration inram pressure stripped tails. While we have discussed somedi ff erences in tail properties as a function of height above thedisk and across the di ff erent simulations, in this section wefocus on the impact of the ICM properties on cloud proper-ties. We find that both the ICM density and velocity influencethe gas in the stripped tail.6.1. ICM Density
First, we compare the LDLV and HDLV simulations to dis-cuss the e ff ect of the ICM density on cloud properties. In Fig-ure 9 we showed that at the same height, the clouds in HDLVare moving slightly faster than those in LDLV, and in Figure4, we see that HDLV has a longer dense tail than LDLV atthe same simulation time. In our model, this follows from the fact that there is more mass in the higher density windresulting in quicker mixing for HDLV clouds.We also find that the maximum number and mass of cloudsin any height bin is larger in the HDLV run than in the LDLVsimulation. While this must, to some extent, reflect the factthat the ram pressure ( ρ v ) is about twice as high in HDLVthan in LDLV, and there is therefore more stripped gas thatcould cool to dense clumps, we argue that that is not thewhole story.If we compare the relative number of clouds as a functionof distance from the galaxy we see a dramatic di ff erence inthe two runs. From Figure 10, we can see that, at their respec-tive peaks, there are about six times as many clouds at 15 kpcfrom the disk as 85 kpc above the disk in LDLV, while thereare about two times as many clouds near the disk when doingthe same comparison for the HDLV run. The di ff erence iseven more dramatic when we compare the number of cloudsat 155 kpc in the two LV runs. The maximum number andmass of clouds in HDLV is larger at 155 kpc than at 85 kpc,in stark contrast to the continued decline in the number andmass in clouds as a function of height in LDLV.In addition, if we track the total dense gas mass within theentire simulation, as in Figure 12, we find that the dense gasmass increases in HDLV and decreases in LDLV.9This suggests that a higher density ICM leads to morecloud condensation due to stripped gas mixing. This is con-sistent with the results of section 5.3 – in particular, when welook closely at Figure 13, we see that LDLV has the largestpopulation of clouds for which t cool , mix ≤ . t cc .Examining the ICM fractions of dense gas in the HDLVand LDLV simulations can give us insight into whether mix-ing or another process such as compression is driving the dif-ferences in cloud mass in HDLV and LDLV. Clouds in theHDLV tail tend to have higher ICM fractions, with the dif-ference becoming more pronounced at larger height from thegalaxy, as seen in both Figures 9 & 11. This again followsour model as more mass from the wind will have impacted acloud in a higher density ICM. The di ff erent ICM densitiesdo not have a dramatic impact on the minimum-to-maximumICM fraction ratio, although there is a slight tendency forclouds in the LDLV run to have closer to uniform ICM frac-tions. Therefore cloud mixing occurs in a similar fashion inboth simulations.Comparing the time evolution of low f ICM gas to densegas in Figure 12 also indicates a significant di ff erence in howmixing e ff ects gas in the stripped tails due to ICM density. Inboth runs the low f ICM gas increases first, then as gas contin-ues to mix with the ICM the low f ICM gas mass decreases. InLDLV the dense gas mass decreases when the low f ICM gasdecreases, indicating that mixing in low density ICM resultsin gas heating. On the other hand, in the higher density ICMof HDLV, low f ICM gas mass decreases while dense gas masscontinues to increase, indicating that mixing in HDLV resultsin gas condensing into clouds.In summary, we argue then that stripped gas in a higherdensity surrounding medium accretes more of the ICM, re-sulting in gas with higher f ICM , and the formation of morehigh-density clouds, particularly at larger distances from thegalaxy. 6.2. ICM velocity
We can now compare HDLV and HDHV to discuss thee ff ects of the ICM wind velocity on the stripped gas. Un-surprisingly, the most clear di ff erence is that stripped gas ismoving faster in the high velocity run.Comparing the number and total mass of clouds as a func-tion of height, we see that although the maximum number ofclouds near the disk in HDHV is higher than in HDLV, as wewould expect since the higher ram pressure in HDHV willstrip more gas, the number drops by a slightly larger factorfrom 15 kpc to 85 kpc in HDHV than in HDLV at their peaks.However, the di ff erence is dramatic at the largest distancewe probe–155 kpc from the disk, there are more clouds inHDLV than in HDHV at their peaks. This indicates that thefast, high-Mach number HDHV wind does not allow dense clouds to survive (or form) as easily as HDLV . Indeed, wealso see this in the mass of dense gas in the wake as a func-tion of time – the mass drops dramatically in HDHV while itincreases in HDLV (Figure 12). This agrees with our findingin Figure 13 that more clouds have shorter t cc than t cool .Interestingly, the gas composing HDHV clouds has a largerange of ICM fractions, particularly close to the galaxy (Fig-ure 11). Perhaps the range in ICM fractions within a cloudis a signature of the clouds being destroyed, as indicated bythe short cloud crushing times. We also find that at the sameICM density the faster wind (HDHV) results in clouds withlower minimum ICM fractions (in other words, less mixingbetween the stripped and surrounding gas). This supports ouranalytic model that the faster ICM will add more momentumto stripped gas per unit mass. In addition, almost all identi-fied clouds in HDHV have minimum f ICM less than 0.5. Incombination with the fact that the mass of low f ICM gas inthe wake is always more than the mass of high density gas(Figure 12), this again highlights that mixing is likely to heatcold, dense gas into the di ff use ICM. DISCUSSION7.1.
The Importance of the ICM
In previous work we have argued that the ICM is impor-tant in setting the properties in the tail, from the X-ray lu-minosity to the star formation rate (Tonnesen et al. 2011;Tonnesen & Bryan 2012). We found that both the X-ray lu-minosity of the stripped tail and the star formation rate instripped gas will increase as the ICM pressure increases dueto the near-pressure equilibrium of the stripped and surround-ing gas. These conclusions were largely based on the ρ -T di-agrams of the stripped gas, which found higher density gas atall temperatures in a higher-pressure ICM.In this paper, we focus on the properties of overdenseclumps in the tail. As we have discussed, we find agree-ment with our previous conclusion that a higher density ICMproduces more high density clouds.Here we look more closely at our clouds to determinewhether they are likely to be gravitationally bound or pres-sure confined. To do this we compare estimates of the cloudinternal energy and the gravitational potential: U = k boltz T mean M cloud µ m H (11) U g = GM r cloud (12)In which T mean is the mass-weighted average temperature,M cloud is the total mass of the cloud, and r cloud is the cloud However, as we do not track individual clouds we cannot discount the pos-sibility that there is a larger range of cloud velocity in HDHV that is spread-ing an increasing number of clouds more thinly across the length of the tail. U g (cid:28) U ). In our higher-density ICMruns (HDLV and HDHV), some of the most massive cloudslie very close to the line of equality. These clouds seem tobe largely pressure confined, which agrees with our earlierfindings of near-pressure equilibrium in the stripped tail andICM, although we cannot rule out a part of these clouds be-ing gravitationally unstable. Indeed, when we select onlyclumps with at least 300 cells, as in Appendix A1, we find amuch higher fraction of massive clouds lie close to the line ofequality, supporting the idea that regions of massive cloudsmay be gravitationally unstable.Because our model for gas acceleration requires the mix-ing of ICM and ISM gas, we have carefully examined themixed fraction of dense clumps. In Figure 11 we saw evi-dence that the surrounding ICM is not just compressing thestripped material towards pressure equilibrium, but is mixedthroughout the stripped gas, even in the most dense clouds.In fact, we see that this mixing is more e ffi cient in the HDLVrun than the LDLV run, indicating that a higher density (andtherefore higher pressure) ICM mixes more with the strippedtail.The mixing of the stripped and surrounding gas can eitherresult in the cooling of hot gas or the heating of cold gas.Clouds tend to lie close to the t cool , mix = t cc line, indicatingthat they may be able to survive for significant amounts oftime. In addition, as we have mentioned in Section 5.3, thedensity ratio of clouds to the ICM ranges from 10 - 10 .Therefore, in a picture in which the mass accretion of denseclouds takes more time for higher density ratios as in GO18(at fixed t cool , mix / t cc ), the cooling onto these clouds could takemore than 1 Gyr (10 cloud crushing times of 10 years).These results lead us to a picture in which stripped gasquickly mixes with the ICM. This happens more quicklywhen the surrounding gas has a higher density (and thereforepressure). We posit that this plays out in di ff erent ways inthe HDLV and HDHV runs. In pressure equilibrium, coldgas will have higher densities so the cooling time will beshorter and clouds will be more likely to survive. We seethis in the HDLV run in the low t cool , mix / t cc clouds in Fig-ure 13 and in the increasing number and mass of clouds from ∼
85 kpc to ∼
155 kpc. However, although the cloud densitiesare similar in the HDHV run, the velocity di ff erence betweenthe cloud and the ICM is also larger, leading to a lower t cc ,and therefore more clouds reside above the t cool , mix = t cc linein Figure 13. Therefore, when stripped gas has mixed with alarge fraction of ICM gas (i.e. f ICM (cid:38)
Figure 14.
The internal energy of clouds versus the gravitationalpotential of clouds. As in Figure 9, the contours are based on thedensity of clouds in each two-dimensional histogram bin. The pan-els are organized as in Figure 9, and have a line at x = y . We seethat the internal energy of the clouds is higher than the gravitationalpotential, indicating that these clouds are pressure confined. destroyed (Figure 12). This is why we do not see clouds withhigh ICM fractions in Figure 11.Therefore the interplay of ICM density and velocity willdetermine whether gas mixing results in dense clouds or dif-fuse gas. 7.2. Making Predictions for observations
Star Formation in Stripped Tails
Although we do not include star formation in these simula-tions and only focus on dense gas, in observations of strippedtails some of the best evidence for cold dense clouds is theHII regions resulting from star formation. However, cur-rently there are no clear cases of stars formed more than ∼ ∼
155 kpc.There are several possible explanations for this. On theobservational side, stars at such a large distance from the diskmay not be as clearly associated with a stripped tail becauseall lower density gas will have been mixed and heated until itis indistinguishable from the ICM.On the simulation side, we first recall that our clouds arepressure confined, and our rough estimate finds that t cool ∼ t cc , so many of our clouds could be destroyed before theywould cool and form stars. Also, it is possible that star for-mation in dense gas closer to the disk will heat nearby gasand cause it to mix into the surrounding medium more e ffi -ciently than we find in our simulations. A highly-resolvedsimulation including star formation is required to determinehow far from the disk stars will form.However, despite these caveats, based on our results wewould predict that star formation is most likely to occur inthe stripped tails of galaxies moving slowly in a high-densityICM. This might translate into more star formation fromstripped galaxies with circular orbits close to the cluster cen-ter. 7.4. Caveats
In this section, we briefly discuss a number of shortcom-ings of our simulations, first discussing the impact of spatialresolution, before turning to other physical e ff ects that wehave not included. 7.4.1. Resolution
Although our resolution of up to ∼
40 pc allows for manycells across the galaxy disk, individual clouds in the tail arenot well-resolved. We require at least 10 cells in a cloud, butin order to begin to resolve an individual cloud, ∼ wouldbe required (32 cells across the cloud radius). However, theindividual clouds we identify are objects in the flow and havealready been processed by turbulence and cooling, while theresolution requirement of clouds in cloud-crushing simula-tions refers to the initial cloud size – such simulations alsogenerally find structure in their partially mixed clouds downto the grid scale. Therefore, we argue that the in-situ cloudsizes are not necessarily a good estimate of the resolved na-ture of the simulations.We examine the impact of resolution in more detail in Ap-pendices A and B, where we look at the properties of cloudsas a function of their resolution as well as an additional set ofsimulations performed at lower resolution. The overall con-clusion is that, while detailed properties do change, the over-all results are remarkably robust to changes in resolution.However, we note that if a cloud does fragment into smallpieces, as in the “shattering" scenario of McCourt et al.(2018), because we are not resolving the c s t cool length scale,the fragments would mix into the surrounding medium at thegrid scale. Our clouds are not in precise pressure balancewith the ICM, perhaps making a shattering scenario more rel-evant (see discussion in Gronke & Oh 2020).Future simulations of clouds surrounded by gas at thesehigh density and temperature ratios are required to predictthe ICM densities and temperatures at which clouds will mixinto the ICM or will accrete gas from the ICM. In this paperwe only claim that higher ICM density is likely to result inlonger cloud survival, while higher relative ICM velocity willresult in faster cloud mixing.7.4.2. Missing Physics
We run these simulations using only hydrodynamic equa-tions, which means that we may be missing relevant physicsfor gas mixing.For example, we have no magnetic fields. Gronke & Oh(2020) and others (e.g. Sur et al. 2014; McCourt et al. 2015;Cottle et al. 2020) include magnetic fields in their simula-tions that study the growth of cold clouds through the en-trainment of the surrounding material. Their magnetic fieldsincrease the velocity of clouds, suppress the KH instability,and stretch the cold gas along the field lines. Importantly,they find that magnetic fields have little impact on the overallmass growth rate. In contrast, in plane-parallel simulations,2Ji et al. (2019) find that magnetic fields suppress cold gasgrowth through entrainment. While it is not clear whethermagnetic fields would suppress or enhance the growth of coldclouds, it is likely that they would stabilize them against mix-ing to some degree.Along with a magnetic field, we do not include ther-mal conduction. Cloud survival has been studied includingisotropic heat conduction. Bruggen & Scannapieco (2016)include both radiative cooling and heat conduction in simu-lations of clouds being ejected in galactic outflows. They findthat while the outer envelope is evaporated, the central regioncools and stretches into dense filaments (in agreement withsimulations that only include radiative cooling: Mellema etal. 2002; Fragile et al. 2004; Orlando et al. 2005; Johansson& Ziegler 2013). These clouds are accelerated only at earlytimes because their cross-section decreases dramatically asthe outer layers evaporate, so they move more slowly thansimulated clouds without heat conduction. Clouds with heatconduction also lose mass more quickly than those with onlyradiative cooling.The authors do highlight that this time can be lengthenedif there are magnetic fields perpendicular to the temperaturegradient (Cowie & McKee 1977; Cox 1979). Vollmer et al.(2001) argue that magnetic fields could increase the evapo-ration time by nearly an order of magnitude, although theactual value is highly uncertain. In previous work, we havefound that in order for the length of tails in our simulationsto agree with observations, heat conduction must not be e ffi -cient (Tonnesen & Bryan 2010; Tonnesen et al. 2011).As with increased resolution, we think that including thesephysical e ff ects will not alter the trends we have found in thispaper. Interestingly, recall that in the HDHV run, the cloudvelocities far from the disk are below our analytic predic-tion. This may be explained by the single-cloud results thatwith radiative cooling clouds stretch into narrower filamentsover time, while our analytic model assumes a constant cloudcross-sectional area. CONCLUSIONSWe have presented an analytic model describing how gas isunbound and accelerated away from a galaxy due to mixing-mediated ram pressure stripping. The model is based on theidea of mixing driving momentum transfer, rather than a tra-ditional ram “force". To verify this model and to highlightthe impact of the ICM velocity and density, we also presentthree “wind-tunnel" simulations of a galaxy undergoing rampressure stripping in which we identify and examine cloudsin the tail. Our main conclusions are as follows:1. We present a model in which the acceleration of gasfrom a galaxy is due to mixing with the ICM, and isbased on a mix of energy and momentum depositioninto galactic gas from the ICM (Section 2). The model makes clear predictions that gas with higher velocitiesand at larger distances from the disk will be more well-mixed with the ICM (Figures 1 & 2).2. We compare our model to three simulations in whichwe have varied the ICM velocity and density. We findexcellent agreement with the ICM fraction-velocity re-lation for all gas in the stripped tail (Section 4.1).3. When we focus only on the dense clouds in our sim-ulations we still find good agreement, even when wefold in a model for the distance a cloud has traveled(Figure 9).4. Clouds are nearly uniformly mixed with the ICM,meaning that in our simulations there is no cloud“core" that survives intact from the galaxy. The ICMfraction in dense clouds increases as a function ofheight from the disk (Figure 11). Comparing simula-tions, we find that at the same wind velocity, clouds aremore well-mixed in a higher density ICM. At the sameICM density, clouds are more well-mixed in a slowerwind.5. Both the number and mass of gas in dense clouds de-creases as a function of height above the disk (Fig-ure 10), suggesting that clouds generally do not survivein the tail. However, in the HDLV run, the number andmass of clouds increases from ∼
85 kpc to ∼
155 kpcabove the disk, indicating that some clouds survive andaccrete gas from their surroundings. Stripped gas mix-ing with the ICM results in decreasing dense gas massin HDHV and LDLV, but in HDLV mixing adds densegas mass to the simulation (Figure 12).Importantly, we have shown in this paper that dense cloudsin stripped tails are part of the continuum of gas in the tails.Our mixing-driven model for gas acceleration in the tail ap-plies equally well to low-density and high-dense gas. Thisis an important departure from the simple picture that in-tact clouds can survive being “pushed" from the galaxy byan ICM wind. Indeed, this leads to the observational pre-diction that the metallicity of dense clouds should decreaseas the distance from the galaxy increases (absent enrichmentdue to additional star formation in the tail).Because mixing drives the formation and acceleration ofthe stripped gas, the question to ask when comparing the dif-ferent temperature and density distributions of tails becomesclear: will mixing with the ICM result in more gas being ableto cool into dense clouds, or will it heat the stripped gas untilit is indistinguishable from its surroundings? We have shownthat a higher density ICM will tip the balance towards moredense cloud formation, while higher relative ICM velocitieswill destroy clouds and add the stripped gas to the di ff usesurroundings.3Our results on cloud survival and mixing are based on com-paring simulations with di ff erent ICM properties, howeverwe also consider cloud survival more theoretically by calcu-lating their crushing and cooling times. We find that cloudsfall near the t cool , mix = t cc line when we use the cooling timeof the mixing layer of our clouds (Figure 13), indicating thatcloud survival cannot be universally predicted even withina single stripped tail (or that the process of cloud evolutiondrives clouds close to this relation).Given the diversity of observed stripped tails, some withdense gas and even star formation, while others only containmore di ff use gas in their tails, the question of what causesdense gas survival and collapse to stars is important to under-standing the physics at play in the ICM. While our results arean important step outlining the competition between velocityand density, predicting the survival of dense gas in strippedtails in detail will require more theoretical work – in partic-ular, a suite of simulations that resolve individual clouds and include star formation as well as non-ideal MHD processessuch as conduction.ACKNOWLEDGMENTSWe would like to thank the referee for helpful com-ments that improved the paper. The authors gratefully ac-knowledge support from the Center for Computational As-trophysics at the Flatiron Institute, which is supported bythe Simons Foundation. ST thanks the GASP collabora-tion for useful discussions about gas acceleration in strippedtails. GLB acknowledges financial support from NSF(grant AST-1615955, OAC-1835509), and NASA (grantNNX15AB20G), and computing support from NSF XSEDE.The simulations used in this work were run on facilities sup-ported by the Scientific Computing Core at the Flatiron Insti-tute, a division of the Simons Foundation.APPENDIX A. CLUMP RESOLUTIONIn this appendix we discuss individual cloud resolution by comparing the properties of clumps that include at least 300 cells toour entire set, which includes clumps with as few as 10 cells.In Figure A1 we repeat Figure 11 using only those clumps with at least 300 cells. The most striking, although unsurprising,di ff erence is that there are many fewer clumps using this selection criteria. We also note that the ratio between the minimum andmaximum ICM fractions within each clump tends towards lower values. Again, because we select larger clumps we would expectthis result. Interestingly, the di ff erences between the ICM fraction minimum-to-maximum ratio as a function of height above thedisk are clearer when we focus on larger clouds. Specifically, clumps identified at 15 kpc above the disk tend to have lower ratios,in other words be less well-mixed, than those at 85 kpc or 155 kpc. This agrees well with our conclusions that mixing is drivingcloud acceleration and that unmixed cores are not surviving to large distances.In Figure A2 we recreate the bottom panels of Figure 13 to determine whether the trends we see for the whole cloud samplecontinue when only considering the largest clouds. As one would expect, when focusing on larger clouds the t cool , mix / t cc decreases.As t cc increases with cloud radius, even at the same cloud density we would expect a lower t cool , mix / t cc , as we see.When we look at only the largest clumps in our tails, we see that the vast majority lie well below the line of equality, indicatingthey will grow in mass rather than destroyed. However, we note that the number, mass, and mass flux of large clouds decreasesfrom 15 kpc to 155 kpc in both LDLV and HDHV.We also note that there still does seem to be a di ff erence in the t cool , mix / t cc distributions in the large clouds, with HDLV valuestending to be lower. B. SIMULATION RESOLUTIONIn this section, we examine the impact of the maximum resolution in the simulations on the evolution of mixed and dense gasin the tails. We have rerun all three simulations using lower resolution. LDLV and HDLV were run with a maximum resolutionof 160 pc, and HDLV was run with a maximum resolution of 320 pc. In Figure B3, we repeat Figure 12 with the lower resolutionsimulations.First, we highlight the striking similarity in the LDLV and HDHV simulations at high (40 pc) and low (160 pc or 320 pc)resolution. At lower resolution, we see that there is always more low f
ICM gas in the simulation than there is dense gas, and thisis reflected in the wake. This indicates that in general, as gas mixes it is heated and di ff uses into the ICM. However, as with thehigher resolution runs, HDLV tells a di ff erent story. Here the mass of dense gas in the wake is larger than the mass of low f ICM gas, indicating that gas mixes and cools. Interestingly, we do not see the growth of high-density gas in the wake as in the higher4
Figure A1.
The same as Figure 11, but only including clumps with at least 300 cells. The same trends hold as in the full clump sample, andwe can clearly see that individual clumps tend to be more uniformly mixed (have minimum-to-maximum ratios closer to unity) farther from thedisk.
Figure A2.
The same as the bottom three panels in Figure 13, but only including clumps with at least 300 cells. These larger clumps tend to liewell below the line of equality.
Figure B3.
As Figure 12, but for simulations run at either 160 pc (LDLV and HDLV) or 320 pc (HDHV) resolution. We see that particularly forLDLV and HDHV the unmixed and dense gas trends are remarkably similar to the 40 pc resolution runs. Although the specifics of HDLV di ff erat the lower resolution, the fact that more dense gas than unmixed gas exists in the wake at late times is consistent with the higher resolutionresults. resolution simulation. We highlight that at lower resolution we find the same qualitative trends–mixing gas does not becomedense in LDLV and HDHV, while mixing gas can become dense in HDLV.REFERENCES Abramson, A., Kenney, J. D. P., Crowl, H. H., et al. 2011, AJ, 141,164Agertz, O., Moore, B., Stadel, J., et al. 2007, MNRAS, 380, 963Begelman, M. C., & Fabian, A. C. 1990, MNRAS, 244, 26P Bellhouse, C., Ja ff é, Y. L., Hau, G. K. T., et al. 2017, ApJ, 844, 49Bellhouse, C., Ja ff é, Y. L., McGee, S. L., et al. 2019, MNRAS,485, 1157 Bryan, G. L., Norman, M. L., O’Shea, B. W., et al. 2014, ApJS,211, 19Brüggen, M., & Scannapieco, E. 2016, ApJ, 822, 31Burkert, A. 1995, ApJL, 447, L25Chandrasekhar, S. 1961, International Series of Monographs onPhysicsCottle, J., Scannapieco, E., Brüggen, M., et al. 2020, ApJ, 892, 59Cowie, L. L., & Songaila, A. 1977, Nature, 266, 501Cowie, L. L., & McKee, C. F. 1977, ApJ, 211, 135Cox, D. P. 1979, ApJ, 234, 863Cramer, W. J., Kenney, J. D. P., Sun, M., et al. 2019, ApJ, 870, 63Crowl, H. H., Kenney, J. D. P., van Gorkom, J. H., et al. 2005, AJ,130, 65Fielding, D. B., Ostriker, E. C., Bryan, G. L., et al. 2020, ApJL,894, L24Fossati, M., Mendel, J. T., Boselli, A., et al. 2018, A&A, 614, A57Fossati, M., Fumagalli, M., Boselli, A., et al. 2016, MNRAS, 455,2028Fragile, P. C., Murray, S. D., Anninos, P., et al. 2004, ApJ, 604, 74Gronke, M., & Oh, S. P. 2020, MNRAS, 492, 1970Gronke, M., & Oh, S. P. 2018, MNRAS, 480, L111Gullieuszik, M., Poggianti, B. M., Moretti, A., et al. 2017, ApJ,846, 27Gunn, J. E., & Gott, J. R. 1972, ApJ, 176, 1Hernquist, L. 1993, ApJS, 86, 389Ja ff é, Y. L., Poggianti, B. M., Moretti, A., et al. 2018, MNRAS,476, 4753Ja ffff