arXiv:2011.00042v1 [astro-ph.GA] 30 Oct 2020 omn ein a ei tt fgoa rvttoa con gravitational global of state a in see that be traction, evidence may numerical regions and forming observational growing is There INTRODUCTION 1 MNRAS ⋆ 2000 Pudritz & Fiege 1956 Bonnor ( 07TeAuthors The 2017 © ofiuain (e.g. th configurations of whic cyl analysis provide stability layers, linear filaments is The self-gravitating respectively. scenario collisio to spheres, of This birth large instability cores. dynamical give of multiple the they tha result into on, fragment revealed evolu- a Later then the has as matter. during diffuse work form of scales structures Theoretical large dense clouds. sufficiently flattened molecular of of range manif tion wide hereafter) (GI a instability in gravitational by that plies shown was collaps bot as hierarchical with medium, the simulated exhibit gas turbulent atomic cooling diffuse and warm self-gravity of flows of lisions .V Zhuravlev V. V. clouds molecular of formation in dust of role Dynamic ouin (e.g. solutions 1 2007 trbr srnmclIsiue oooo ocwState Moscow Lomonosov Institute, Astronomical Sternberg -al [email protected] E-mail: and ) 000 , aaj-oeoe al. et Naranjo-Romero 1 ´ zuzSmdn tal. V´azquez-Semadeni et ; – asn1969 Larson ?? lere lere 1978 Elmegreen & Elmegreen (2017) eox1951 Ledoux swl stecrepnignon-linear corresponding the as well as ) 1 ⋆ ; eso 1969 Penston ; hnrska em 1953 Fermi & Chandrasekhar ( h rf fgan stknit con,te ei growing begin they account, into taken ( is grains of drift the ftemdu nscales considerabl may on gas medium the the to of respect with grains dust of motion fteuiomsl-rvttn a trs r agnlys marginally are rest at gas self-gravitating uniform the of h cloud, the e words: Key stages different at size interste evolution. sub-Jeans cold s of domains of of dusty absence contraction produce the gravitational th in facilitate larger mixture substantially can gas-dust is of rate growth instability the resonant and wave well the as onto projected nature velocity drift the with propagating on rate otato ftedfuegsit oeua lusi comm is fraction, clouds mass molecular small into the of gas because diffuse the of contraction ut xicin—sas omto tr:pootr pr — protostars stars: — formation stars: — extinction dust, pertur gas-dust growing the dust, the du of the drift of bulk perturbations subsonic by caused attraction gravitational ABSTRACT 2015 fτ ( 2019 ) uti h sa io opnn fteitrtla medium interstellar the of component minor usual the is Dust o xml.Ti im- This example. for ) ´ zuzSmdn tal. V´azquez-Semadeni et 1 / ; 3 .Tesproi col- supersonic The ). auaa&Inutsuka & Masunaga f t ff − ; stersnneo ev on ae tpe yteself-gra the by stopped waves sound heavy of resonance the is t 1 ff aaaa1987 Nagasawa where , rvtto yrdnmc ntblte ae S:c ISM: — waves — instabilities — hydrodynamics — gravitation h hsclmcaimrsosbefrsc ekdepende weak a such for responsible mechanism physical The . s idealised ese nvriy nvriesi r,1,Mso 124 Russi 119234, Moscow 13, pr., Universitetskij University, nesand inders ssitself ests fthe of e τ stesopn ieo risepesdi nt ftefe f free the of units in expressed grains of time stopping the is rpitNvme ,22 oplduigMRSL MNRAS using Compiled 2020 3, November Preprint by d star λ ns h h - t ; ; . λ J where , cleouino h a osa omto ( formation star to way an the thermal its on evolution for agent ical an as considered usually hereafter) fraction time. the fall commonly free inverse are configurations realistic gr of largest GI The for scale. large f infinitely inverse the the at to manifests (corresponding time) Jean rate the growth to largest contrast the in value when finite a fo has c scale configurations the istic unstable for t most appropriately the includes However, chosen which configuration. density sponding scale, and sound Jeans of the speed to cal wave similar static always configurations damping are realistic and of above GI growing for scales the critical the are Thus, scal length with Jeans Perturbations the v length. Jeans than the the approaches as wavelength to referred the subs now the as at vanishes propagating which waves velocity, sound h heavy background are a pressure such finite on perturbations plane-wave that lished by noted was as time, same the At view. this confirm others) ( 2000 atrcs,tesuyo Ige akto back th goes In GI medium. of uniform study unbounded the the case, mu for latter differ derived not one does basic which the condition, instability the for cial 1985 f uti opnn ftedfueitrtla eim(ISM medium interstellar diffuse the of component a is Dust ; ≃ ,teseicgoer fsl-rvttn bet sntc not is objects self-gravitating of geometry specific the ), iaae l 1987 al. et Miyama λ 0 J . 01 steJaslnt-cl.Telna perturbations linear The length-scale. Jeans the is oee,a hw nti td,tecollective the study, this in shown as However, . lrgsit lusadadditionally and clouds into gas llar ; tfato.Oc hr sstationary is there Once fraction. st etr hi rwhhsaresonant a has growth Their vector. ntua&Mym 1997 Miyama & Inutsuka nta ftercnl discovered recently the of that an ain at bations fmlclrcodfrainand formation cloud molecular of al at table l-rvt.Tenwinstabilities new The elf-gravity. otiuet h destabilisation the to contribute y tpaeaydiscs otoplanetary tart prxmtl qa to equal approximately rate a at nyasmdt enegligible be to assumed only a t yai oei the in role dynamic Its . λ λ < λ Jeans ≃ iyo a ihweak with gas of vity iihdse l 2020 al. et Girichidis λ J ( c ftegrowth the of nce J 1902 u sso as soon as but , A T eoewaves become od ISM: — louds E tl l v3.0 file style X ,woestab- who ), l ieof time all mentioned n many and whrates owth chem- d hfrom ch result, s larger e fthe of s e fall ree Larson real- r aving orre- onic alue ypi- ru- s. e ; 2 V.V. 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Krause et al. 2020). The measured mass fraction of dust with re- see Hopkins & Squire (2018b)(HS18 hereafter), which may be rel- spect to gas in the Milky Way is around 0.01 (Draine 2011). It evant in the neutral circumstellar medium. For the non-linear out- might seem that such a small value rules out the possibility that come of this particular instability see Moseley et al. (2019). It is the dust could dynamically affect the formation of dense clouds important that the resonant instability of the gas-dust mixture is of neutral/molecular hydrogen or even the subsequent collapse of characterised by a weak dependence on the dust fraction. Its growth prestellar cores. rate usually scales as the square or even the cube root of the dust Until recently, dust has been considered as only a passive con- fraction. At least for the particular model of the dust streaming in stituent of the clouds which, however, could be only partially cou- protoplanetary disc, this feature was explained by the mode cou- pled to the gas for sufficiently large grains. This feature may lead to pling of gas-dust perturbations, see Zhuravlev (2019). This implies concentration of dust. Indeed, grains dynamically interact with the that the resonant instability may be important in application to the gas due to the aerodynamic drag (Whipple 1972; Weidenschilling ISM, where the dust fraction is typically small. Furthermore, it may 1977) parametrised by the characteristic stopping time, which is not only provide the dust clumping but also significantly affect the the time over which a particular grain loses its initial velocity in gas dynamics. the absence of other forces. As the stopping time becomes longer, This work is concerned with GI of the partially coupled gas- grains may gain higher velocity relative to the gas. First, station- dust mixture taking into account gas and dust aerodynamical inter- ary bulk drift of the grains under the action of the anisotropic in- action. The linear stability analysis of an unbounded uniform self- terstellar radiation field may occur. This is produced by the ra- gravitating medium is carried out in the two-fluid approximation diation pressure force along with photoelectric and photodesorp- with dust assumed to be a pressureless fluid. Hence, the objective tion forces, see Weingartner & Draine (2001). They show that suf- of this study is to generalise the classical plane wave solution ob- ficiently large grains, up to the micron size, experience consid- tained by Jeans for the dynamics of two partially coupled fluids. erable subsonic drift in the warm and cold ISM. This effect can It is shown that in this case the gas-dust mixture is unstable at all be enhanced up to the transonic and even supersonic drift in the scales. Additionally, the dust is allowed to drift through the gas un- vicinity of bright sources such as AGN and starburst regions. Next, der the action of some external force. In the latter case, this study the dust sinks down to pressure maxima. This feature is widely generalises the HS18 model. The resonant instability of a new type known in the context of dust dynamics in protoplanetary discs, is found at the (sub-)Jeans scale. As far as the drift velocity is suf- as it causes a global inward radial drift and vertical settling of ficiently small (or equal to zero), this instability operates due to the solids along with their local concentration in axisymmetric pres- dust back-reaction on gas arising from the dust self-gravity. If the sure bumps/zonal flows or in the long-living vortices generated by drift velocity is higher than some critical value, the instability op- the turbulence (Johansen et al. 2014). Dust sedimentation in the po- erates due to the known aerodynamical dust back-reaction on gas tential well of an interstellar gas cloud in hydrostatic equilibrium is caused by the bulk drift of the dust subject to external force. In another example of the dust drift considered by Flannery & Krook the latter case, the instability is more prominent than that of HS18 (1978). It was shown that micron-sized grains settle to the centre for the subsonic drift. The growth rate of the new instabilities de- of a cold uniform cloud at a characteristic time not much exceed- pends on either the square root or the cube root of the dust fraction, ing the free fall time of the cloud. In the past few years the rela- which is defined by the different critical value of the drift velocity. tive motion of dust in turbulent clouds has been studied employing It is stated that the new resonant instabilities can affect the grav- the numerical simulations, see Hopkins & Lee (2016), Lee et al. itational collapse of various dust-laden objects, where grains are (2017b), Tricco et al. (2017), Monceau-Baroux & Keppens (2017) significantly decoupled from gas. and Mattsson et al. (2019). These studies revealed the significant A related problem has been studied in protoplanetary discs in fluctuations of the dust density of (sub-)micron-sized grains at sub- the context of planetesimal formation. A dense sub-disc of macro- parsec scales, though there remains a discrepancy in the magnitude scopic solids having small but non-zero velocity dispersion is em- of overdensities obtained by various numerical methods. In contrast bedded in a gas disc, which can usually be assumed gravitation- to similar problem in protoplanetary discs, turbulence in molecu- ally stable. In the absence of aerodynamic drag, which damps lar clouds is supersonic, which complicates the underlying physics. the velocity dispersion of solids, the sub-disc would be gravita- The dust is dragged by the compressible gas, which implies that tionally stable as well. However, taking into account aerodynamic the dust clumping is additionally produced by the compression of drag makes the sub-disc of solids unstable. The corresponding in- the gas-dust mixture, as well as by fluctuations of the drag itself. stability is referred to as the secular GI, which operates when Note that currently turbulent concentration of interstellar dust has both the relative motion of gas and dust and the self-gravity of not been not studied in self-gravitating configurations. Also, no one dust are taken into account, see Youdin (2005) and Youdin (2011) has considered how the dust back-reaction on gas affects the con- who addressed the problem without dust back-reaction on gas, or centration of dust due to the externally driven turbulence. Takahashi & Inutsuka (2014) and Latter & Rosca (2017) who took At the same time, Squire & Hopkins (2018b) recognised a into account the aerodynamic interaction of gas and dust. class of resonant dynamical instabilities inherent in the partially coupled gas-dust mixture when gas and dust interact with each other via aerodynamic drag. For resonant instability to operate, the phase speed of some wave existing in the gas should match the 2 DYNAMICS OF SELF-GRAVITATING DUST-LADEN projection of the drift velocity of dust onto the wavevector, thus, MEDIUM generally dust must flow through the gas. The resonant instabil- 2.1 Two-fluid equations ity can manifest itself in various objects, e.g. in protoplanetary discs (Squire & Hopkins 2018a) or hot magnetised circumstellar Dynamics of the gas-dust mixture can be considered in the two- medium such as stellar coronae and HII regions (Hopkins & Squire fluid approximation. The fluid associated with gas has velocity Ug , 2018a). As the dust drift becomes nearly sonic or even supersonic, while Up is the velocity of the fluid associated with dust. The rela- the most basic case of the acoustic resonant instability is realised, tive velocity of dust with respect to the gas, V Up Ug, drives ≡ − MNRAS 000, 1–?? (2017) Gravitational instability of gas-dust mixture 3 the aerodynamic drag, which couples the two fluids to each other. The corresponding equations of motion and mass conservation are ρ = const, ρ = const, (10) the following g p where the constant dust fraction is introduced as ∂Ug p ρp V +(Ug )Ug = ∇ Φ+ , (1) ρp ∂t ·∇ − ρg −∇ ρg ts f . (11) ≡ ρg Equations (6-10) represent the homogeneous self-gravitating ∂ρg + (ρgUg) = 0 (2) medium, which consists of gas in hydrostatic equilibrium and dust ∂t ∇ · drifting through the gas if it is externally forced. for gas with mass density ρg and

∂Up V +(Up )Up = a Φ , (3) 2.3 Equations for linear gas-dust perturbations ∂t ·∇ −∇ − ts Let the Eulerian perturbations of enthalpy and dust density be de- ′ ′ noted as h and ρp, respectively, while the Eulerian perturbation of ∂ρp + (ρpUp) = 0. (4) gas density be ∂t ∇ · h′ for dust with mass density ρp. Dust is subject to an external force, ′ ρg = ρg 2 . (12) which causes an acceleration a, see equation (3). Aerodynamic cs drag is represented by the last term on the right-hand side (RHS) The Eulerian perturbation of the dust fraction reads of both equations (1) and (3). It is parametrised by the grain’s stop- ′ ′ ′ ρp ρg ping time, ts, which is assumed to be constant in this study. Note f f . that the latter assumption is oversimplifying, since the variations ≡ ρg − ρg of gas density invoke the corresponding variations of aerodynamic As the both fluids are compressible, the relative perturbation of the drag, see HS181. The gas pressure is denoted by p, while dust is dust fraction becomes a meaningful quantity expressed as considered to be pressureless. New to this work is that the mixture ′ ′ ′ f ρp h flows in its own gravitational potential, Φ, which is determined by δ = 2 . (13) ≡ f ρp − c its total density according to Poisson equation s Note that as cs , δ tends to the relative perturbation of dust 2 → ∞ Φ = 4πG(ρg + ρp). (5) density used, e.g., by Zhuravlev (2019). Further, the Eulerian per- ∇ turbations of gravitational potential, gas and dust velocities are, re- In what follows, it is assumed for simplicity that gas is barotropic, ′ 2 spectively, Φ , u and u . The Eulerian perturbation of the relative p = p(ρ ), and, accordingly, p = c ρ , where c is the sound g p g s g s velocity is speed. Thus, the specific pressure∇ gradient∇ in equation (1) can be replaced by the gradient of a new quantity, h p/ρg, where h v = up ug . (14) is equivalent to enthalpy in the particular case∇ of≡∇ homentropic flow. − Equations (1-5) specify the dynamics of a self-gravitating partially Equations (1-5) linearised on the background (6-10) are the coupled gas-dust mixture. following

∂u v V g = h′ Φ′ + f + δ , (15) 2.2 Stationary self-gravitating configuration ∂t −∇ −∇ ts ts   In order to construct the homogeneous stationary solution, the ′ Jeans swindle is expanded here onto the two-fluid model. Thus, ∂ρg + ρg ug = 0, (16) it is assumed that stationary gravitational potential is zero, while ∂t ∇ · the validity of the Poisson equation is guaranteed by some external source of gravity, see e.g. Binney & Tremaine (1987). In this case, ∂up v ′ the stationary solution obeying equations (1-5) is as follows +(V )up = Φ , (17) ∂t ·∇ − ts −∇ Φ = 0, (6) ′ ∂ρp ′ +(V )ρ + ρp up = 0, (18) ∂t ·∇ p ∇ · Ug = 0, (7) 2 Φ′ = 4πG ρ′ + ρ′ . (19) ∇ g p

Up = V = tsa, (8) They are supplemented by equations (12), (13
) and (14). Taking the divergence of equations (15) and (17) and com- bining equation (16) with equation (18) one arrives at the more p compact set of equations ∇ = f a, (9) ρg 2 ′ ′ 1 ∂ h 2 2 h = h′ + ω (1 + f) + fδ c2 ∂t2 ∇ ff c2 − 1 It can be checked that the addition of the corresponding terms in equa- s  s  f tions given below does not invalidate the basic estimates and conclusions [ v +(V )δ] , (20) made for the resonant instabilities. ts ∇ · ·∇

MNRAS 000, 1–?? (2017) 4 V.V. Zhuravlev

restrictions (24-25) are valid, the relative velocity of the dust in the perturbed motion, v, is not difficult to exclude from equations ∂δ 1 ′ +(V )δ = 2 (V )h v, (21) (20-22). However, the analytical results describing the resonant in- ∂t ·∇ − cs ·∇ − ∇ · stabilities can be obtained with no restrictions on tev and lev, see ∂( v) 1 ∂2h′ 1 ∂h′ Sections 5.1 and 5.2. ∇ · +(V ) ( v) = ∂t − c2 ∂t2 ·∇ ∇ · − c2 ∂t s  s  ′ 1 2 h 2.4 Units v ωff (1 + f) 2 + fδ , (22) − ts ∇ · − c  s  For the problem considered in this study, the natural units to mea- where sure time and velocity are, respectively, ts and cs. Accordingly, the 1/2 length has to be measured in units of csts. In the dimensionless ωff (4πGρg) (23) ≡ version of equations (20-22) the matter’s own gravity is introduced is the inverse characteristic free fall time of the gas component, tff . by the dimensionless parameter The set of equations (20-22) is closed with respect to the (26) scalars h′, δ and v. It describes the divergence of the velocity τ ωff ts, ∇ · ≡ fields of dust and gas leaving its vortex components undetermined. which provides the ratio of stopping time to free-fall time. This Gravity does not affect the vortex components of ug,p within the parameter takes a wide range of values less than unity in molecular model considered in this work. The vortical dynamics of the gas- clouds, see Section 6. dust mixture is non-trivial due to the aerodynamic drag and, though omitted here, is worthy of a separate study. In what follows, ug,p are assumed to be the potential fields. 2.5 Dispersion equation When f 0, equation (20) describes the propagation of → The particular solution in the form of a wave with the dimensionless sound in a self-gravitating gas environment. The second terms f −1 ∝ complex frequency ω measured in units of ts and wavevector k in the first and the second square brackets on the RHS of equation −1 measured in units of (csts) reads (20) introduce, respectively, the gravitational and the aerodynamic ′ ′ feedback from the dust. It may seem that these additional terms can h , δ, v = h¯ , δ,¯ ¯v exp( iωt∗ + ik r∗), (27) ∇ · { ∇ · } − · hardly change the dynamics of ISM, where typically f 0.01, where the dimensionless time and length are, respectively, however, the case is more interesting as heavy sound waves∼ may come into resonance with the trivial dust mode, see Sections 5.1 t t∗ , and 5.2. ≡ ts Equation (21) describes the dynamics of dust in terms of the r evolution of the dust to gas density ratio. The conservation of the r∗ . perturbed dust fraction along the bulk stream of the dust described ≡ csts by the left-hand side (LHS) of this equation is violated through the Equations (20-22) then yield the dispersion equation, which can be terms on its RHS. The first term on the RHS of equation (21) takes expressed as into account the change of the dust fraction due to the bulk advec- Dg(ω, k) Dp(ω, k)= ǫ(ω, k), (28) tion of dust with a certain density into the regions with a different · density of gas. The second term on the RHS of equation (21) intro- with duces the change of the dust fraction due to the divergence of the 2 2 2 2 ω(ω k Vˆ )+ τ (1 + f) perturbed dust motion with respect to the gas. It is this latter term Dg(ω, k) ω k +τ (1+f)+f − · , ≡ − 1 iω + ik Vˆ that provides the clumping of dust in the domains of high pressure − · (29) widely studied in protoplanetary discs. It is determined by equation (22), which is essentially a different form of equation (17). ifτ 2 Accordingly, equation (22) can be thought of as an equation of Dp(ω, k) ω k Vˆ , (30) the perturbed motion of grains considered in the frame comoving ≡ − · − 1 iω + ik Vˆ ′ − · with the perturbed gas. In this case, the terms h on the LHS and of this equation represent the inertial force acting∼ on grains in this iω(ω k Vˆ ) + iτ 2(1 + f) frame. The acceleration of grains with respect to gas is expressed by ǫ(ω, k) f k Vˆ + − · the rest of the terms on the LHS of equation (22), while the terms on ≡ " · 1 iω + ik Vˆ # − · its RHS describe the aerodynamic drag and the gravitational force. 2 f In the limit of the long evolution time of the mixture, ik Vˆ τ 1+ , (31) · − 1 iω + ik Vˆ   − ·  tev ts, (24) ≫ where Vˆ is the drift velocity of dust measured in units of cs. It is as well as the long length-scale of perturbations, assumed that Vˆ < 1 hereafter. Equation (28) accurately determines the linear modes of gas- lev csts, (25) ≫ dust perturbations on the homogeneous background specified by the second term on the LHS of equation (22) becomes the leading equations (6-10). These modes constitute oscillations in the gas- one. It represents the effective gravity acting on grains due to the dust medium at the frequency given by the real part of ω. The acceleration of the gas. When applied to the non-self-gravitating modes can be damping or growing with the amplitude changing gas-dust mixture, the limit (24-25) is known as the terminal veloc- exponentially with time at a rate given by the imaginary part of ω. ity approximation (Youdin & Goodman 2005; Jacquet et al. 2011): The parameters f and τ can take any finite values. However, in the the aerodynamic drag is balanced by the pressure gradient. As the ISM commonly f 1. In this case, the solution of equation (28) ≪ MNRAS 000, 1–?? (2017) Gravitational instability of gas-dust mixture 5

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ˆ Figure 1. The largest ℑ[ω] > 0 among the modes being the solution of equation (28) vs. the absolute value of wavenumber for f = 0.01. V|| = 0.0, 0.1, 0.5, 0.8, 0.9, 0.95 in panels (a), (b), (c), (d), (e), (f), respectively. Solid (black), dotted (red), dashed (green), dot-dashed (blue) and dot-dot-dashed (magenta) lines show τ = 0, 0.01, 0.1, 1.0, 10, respectively. must be close to the basic one corresponding to f = 0. Strictly at referred to as the modes akin to HSW or SDW. The rest of the paper f = 0, equation (28) splits into two independent equations deals with an accurate numerical solution of equation (28) followed 2 2 2 2 by the analytical consideration of the particular situations caused ω = k τ ω (32) − ≡ s by the resonance between HSW and SDW, which takes place at the and mode crossing.

ω = k Vˆ ωp. (33) · ≡ Equations (32) and (33) describe, respectively, perturbations exist- 3 PROFILES OF THE GROWTH RATE ing in gas and a trivial dust mode associated with perturbations of the dust density. The goal of this section is to reproduce the overall picture of GI Equation (32) describes the Jeans instability. As far as k < of a uniform dust-laden medium under the interplay of two effects: τ, the corresponding modes are the two static waves growing and the contribution of the dust to the gravitational potential and the damping at a rate ωs , where k is the absolute value of the mode bulk drift of the dust with respect to the gas. For that, equation (28) | | wavenumber. In contrast, as k>τ, the modes become two waves is numerically solved as an algebraic quintic equation with respect propagating in the opposite directions, which are the heavy sound to ω. According to equation (27), the growth rate of the gas-dust waves (HSW hereafter). The marginal value k = τ kJ specifies mode is represented by [ω] > 0. The largest growth rate vs. the the Jeans length. ≡ absolute value of wavenumberℑ is shown in Figure 1 for the usual Equation (33) introduces a wave of perturbations of the dust value of the dust fraction, f = 0.01 and various τ. density advected by the bulk drift of the dust. Indeed, this equation The very first panel, see Figure 1, represents the special case of can be obtained from equations (20-22)in the limit f 0 provided the dust suspended in the gas, i.e. no bulk drift of the dust, V = 0. ′ → that additionally h 0, which implies that δ tends to the relative Note that physically this means the absence of an external force act- perturbation of the→ dust density. This wave is essentially the one ing on grains, rather than the rigid coupling of the grains with the introduced by Zhuravlev (2019) for the description of the instability gas, since ts is implied to be a finite unit of time throughout this of a gas-dust mixture in protoplanetary discs and referred to as the study. Thus, the relative motion of the grains with respect to the streaming dust wave (SDW hereafter). This term will be used in gas occurs in the perturbed flow. In this case, the self-gravitating this study. The SDW phase velocity equals the bulk drift velocity medium becomes unstable at all length-scales in contrast to the of the dust projected onto k. In the particular case of the suspended well-known Jeans solution. Perturbations with length-scale equal dust, i.e. Vˆ = 0, SDW degenerates into static perturbations of the and smaller than the Jeans length, k>kJ , are no longer stable. dust density. In this study, it is considered along with the general This occurs irrespective of the value of τ, though the growth rate case Vˆ > 0. in the limit of high k decreases visually like τ 2 as one proceeds As the dust fraction acquires a small but non-zero value, the to smaller grains / weaker gravity. For a particular τ, the growth modes of gas-dust perturbations deviate from the solutions of equa- rate approaches that of the Jeans instability at k 0, but retains → tions (32) and (33) discussed just above. Foremost, this occurs due a considerable non-zero value as k kJ , where the Jeans insta- → to the RHS of equation (28), which will be referred to as the cou- bility ceases. In the limit k it reaches a smaller horizontal pling term hereafter. Additionally, there are corrections to the LHS asymptotics. → ∞ of equations (32) and (33) proportional to f. Such modes will be The rest of the panels in Figure 1 represent how the profiles

MNRAS 000, 1–?? (2017) 6 V.V. Zhuravlev

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Figure 2. The curves in top and bottom panels show, respectively, ℜ[ω] and ℑ[ω], where ω is the solution of equation (28). Solid (black) and dotted (red) lines show, respectively, two HSW and SDW obtained for f = 0. Short-dashed (green) and long-dashed (blue) lines show two damping modes, while the dot-dashed (magenta) lines show the growing mode for f = 0.01. The dot-dashed (magenta) line in the top-left panel is not shown as it virtually coincides ˆ ˆ with the dotted (red) line. V|| = 0.0 and V|| = 0.1 in the left and in the right panels, respectively. For all panels τ = 1.0. The filled circles in the bottom-left and bottom-right panels represent the analytical solutions given by equations (45) and (43), respectively. The filled squares represent the analytical solution given by equation (47). of the growth rate change as the bulk drift of the dust progressively found for τ 1, see the particular case in panel (d) in Figure 1. ∼ increases. The growth rate in each panel in Figure 1 is defined by The bumps that emerge at small τ 1 approximately follow the ≪ the projection of Vˆ onto k, which is denoted as Vˆ , so that k Vˆ enhanced profile of HS18 instability for the dust drift getting tran- || · ≡ Vˆ||k. Hereafter, Vˆ|| will be referred to as simply the drift velocity. sonic (see the particular case of τ = 0.1 in panels (d), (e), (f) in It will be assumed to be a positive value. Figure 1). This is so even though they exceed the HS18 instability ˆ As the drift velocity is highly subsonic, the instability of HS18 growth rates as V|| is far from unity (see τ = 0.1 in panel (c) in corresponding to their case of emerges in the range of Figure 1). At the same time, the bumps found at τ 1 exhibit a ts = const ∼ small wavenumbers, k 1, see the solid line in panel (b) in Fig- substantially higher growth rate than that of the HS18 instability ≪ ˆ ure 1 corresponding to τ = 0. As was discussed by HS18, this up to V|| very close to unity (see the particular case of τ = 1.0 in instability appears when the dust drifts through the pressureless en- panel (f) in Figure 1). On the other hand, the bumps corresponding vironment (e.g. cold gas) subject to the aerodynamic back-reaction to τ 1 hardly exceed the growth rate obtained in the absence of ≫ ˆ of dust. The physics of this instability is discussed here in the Ap- the dust drift at the same scales regardless of the value of V||. In- pendix C. As long as Vˆ|| 1, the HS18 instability is weak, so its stead, the dips make the main difference to the growth rate profiles ≪ ˆ long-wavelength side overlaps the Jeans instability already at the for τ 1 as compared to the case of V|| = 0, see panels (e) and ≫ very small τ. However, it dominates at k>kJ for τ . Vˆ||. Ac- (f) in Figure 1. cordingly, the growth of perturbations ceases at high k for τ . Vˆ||. It should be also noted for this case (see the dotted curve in panel (b) in Figure 1) that the growth rate at k kJ has increased as ≃ 4 DESCRIPTION OF MODES compared to the case Vˆ|| = 0. On the other hand, the growth of perturbations for τ & Vˆ|| including the case of τ > 1 remains The particular case of τ = 1.0 is adopted in this section to show the unchanged at all scales. behaviour of the three essential modes of gas-dust perturbations in For a drift velocity approaching the speed of sound, modifica- detail. These modes are the two oppositely propagating HSW and tion of the growth rate profile occurs at all τ. Additional bumps on SDW in the limit of negligible dust fraction. In the same limit, the profiles of the growth rate can be seen in panels (c) and (d) in Fig- remaining two roots of equation (28) are identical to each other rep- ure 1. These bumps are located at sub-Jeans scales, k & kJ . While resenting trivial modes associated with the arbitrary relative motion the dust drift becomes transonic, Vˆ 1, the bumps shift slowly of dust with respect to gas. This motion is damped due to aerody- || → to higher wavenumbers as compared to kJ , compare panels (c), (d), namic drag, thus [ω] 1, and advected by the bulk drift of the ℑ →− (e) and (f) in Figure 1 with each other. At the same time, dips in the dust, thus [ω] kVˆ||. Note that [ω] is referred to as the fre- growth rate on scales between that of the corresponding bumps and quency of theℜ mode→ hereafter, whichℜ is not to be confused with the the Jeans scale get slightly wider. The most prominent bumps are complex frequency in the particular solution (27). For finite f > 0,

MNRAS 000, 1–?? (2017) Gravitational instability of gas-dust mixture 7 the trivial modes are modified by gas dynamics, which, in turn, is 1 induced by the dust back-reaction on the gas. It is checked that the 0,8 non-zero f 1 makes the trivial modes slightly different from 0,6 each other, however,≪ they always remain damped and close to the basic solution for f 0. 0,4

→ Re ¤ At first, each Figure from 2 up to 5 shows the frequencies and 0,2 the growth (damping) rates of HSW having positive and negative 0 frequencies at k>kJ and SDW as the solutions of equation (28) for f = 0. Also, these Figures show the modes of gas-dust pertur- -0,2 bations akin to HSW and SDW as the solutions of equation (28) for -0,4 f > 0. -0,6

4.1 GI on account of suspended dust 0,6 In the absence of the dust drift in the background solution, SDW ex- 0,4 ists in the form of static perturbations of the dust fraction, thus, it is 0,2 represented by the zero frequency and growth rate, see the left pan-

Im ¥ els in Figure 2. SDW is crossed by either of the two HSW branches 0 at the same point k = kJ , see the top-left panel in Figure 2, where -0,2 the both their frequency and their growth (damping) rate vanish as well. As soon as the dust fraction takes a finite value, there are -0,4 the modes of gas-dust perturbations, which are the slowly propa- -0,6 gating damping waves at k = kJ . For k>kJ , these modes ac- quire the increasing opposite frequencies approaching the frequen- 0,8 0,9 1 1,1 1,2 1,3 k cies of HSW, though their damping rate decreases. On the contrary, at the scales longer than some scale corresponding to kkJ . It is seen that the negative frequency mode crossing considered here provides the enhancement of both mode akin to HSW takes a small negative frequency at all k

MNRAS 000, 1–?? (2017) 8 V.V. Zhuravlev

2

1,6

1,2

Re ¦ 0,8

0,4

0

-0,4

-0,8

0,4

0,2

Im § 0

-0,2

-0,4

0,8 1 1,2 1,4 1,6 1,8 0,8 1 1,2 1,4 1,6 1,8 2 k k

ˆ ˆ Figure 4. The same as in Figure 2 for V|| = 0.7 and V|| = 0.704 in the left and in the right panels, respectively. There is a band where growing and damping modes flip over in the bottom-right panel. The filled squares represent the analytical solution given by equation (47). of the mode akin to SDW taken at the Jeans scale decreases as com- 2 ˆ pared to the case of V|| = 0. Additionally, it can be checked that the 1,6 growth of the positive frequency mode akin to HSW gradually ex- 3 pands from k 0 to smaller scales as the drift velocity increases . 1,2 Finally, there are→ no qualitative changes in frequency and damping 0,8 rate profiles of negative frequency HSW, compare Figures 2 and 3.

Re ¨ Further approach to the transonic dust drift, Vˆ 1, causes a 0,4 || → new effect. As soon as the drift velocity increases up to Vˆ|| 0.7 ≃ 0 particularly for τ = 1.0, the mode crossing of positive frequency

HSW and SDW is replaced by what is referred to as the mode cou- -0,4 pling rather than the avoided crossing: compare the top-left panel in Figure 4 and the top panel in Figure 3. The top-left panel in -0,8 Figure 4 shows that the frequencies of modes akin to those cross- 0,4 ing for f 0 become identical to each other in some band of → wavenumbers around the mode crossing scale as the dust fraction 0,2 is finite. This is a known feature of mode coupling, see e.g. the re- view by Fabrikant et al. (1998) and the bibliography referenced by

Im © Zhuravlev (2019). The gas-dust mixture is most unstable inside the 0 band of the mode coupling, whereas the growth of perturbations outside of this band is relatively weak: compare the left panels in Figure 4. Hence, perturbations at the Jeans length-scale, as well as -0,2 in some range of the shorter scales, exhibit much weaker growth than at the scale of the mode crossing. -0,4 Additionally, the right panels in Figure 4 presented for the drift 0,8 1 1,2 1,4 1,6 1,8 2 velocity close to that in the left panels of this Figure show one more k effect. This is the interchange of modes akin to those participating in the mode crossing by the growth/damping rates. It can be seen ˆ that as the drift velocity takes a tiny increase from V|| = 0.7 to ˆ Figure 5. The same as in Figure 3 for V|| = 0.8. Vˆ|| = 0.704 particularly for τ = 1.0, the additional range of scales within the band of the mode coupling appears, where the damping mode turns into the growing one and vice versa, see the bottom- right panel in Figure 4. This interchange by the growth/damping rate always emerges around the crossing scale and expands to both small and large scale as the drift velocity increases. The mode in- 3 Not seen in the Figure 3. terchange stops as it attains approximately k kJ and one finds ≃ MNRAS 000, 1–?? (2017) Gravitational instability of gas-dust mixture 9 the new pattern of modes after the mode coupling is transformed to ∆ is determined by equation the avoided crossing back again, which is demonstrated in Figure 1/2 ˆ ǫc 5 for V|| = 0.8. The frequency of the growing mode, which is re- ∆ . (40) ≈± 2ωc ferred to as the mode akin to SDW as before, has now a different   asymptotics far from the mode crossing. As k 0 and k , Equations (38) and (40) represent the emergence of the res- → → ∞ it approaches the frequencies of SDW and the positive frequency onant instabilities in the vicinity of mode crossings of three and

HSW, respectively, see the top panel in Figure 5. The growth rate two modes, respectively. As soon as τ > 0 along with Vˆ|| > 0, of this mode ceases rather than approaching the free fall rate for there is always a crossing of two modes, which are SDW and HSW k 0: compare the bottom panels in Figures 5 and 3. The second propagating along the same direction. They come into resonance at → mode participating in the avoided crossing in Figure 5 approaches any non-zero f > 0. However, the third mode being the counter- both the frequency and the growth rate of HSW as k 0, while propagating HSW joins the resonance provided that it’s frequency → it becomes damping at sufficiently high wavenumbers. The third is close enough to ωc, or equivalently, the coupling term is suffi- mode akin to the negative frequency HSW has a form qualitatively ciently strong. Depending on the value of Vˆ , both regimes of the similar to that before and remains damping at all wavenumbers: modal resonance just introduced above may occur is astrophysical compare Figures 5 and 3. conditions, where the dust fraction is commonly a small value. While the drift velocity enters the transonic regime, Vˆ|| The location of the mode crossing determined by equation →4 1, the band of instability corresponding to the avoided crossing (34) yields widens. It eventually recovers the growth rate produced by HS18 ˆ instability. It can be noticed that, as the drift velocity tends to unity, ωc = kcV||, (41) the growth rate profile of HS18 instability rises up to the value of where the growth rate found for self-gravitating dust-laden medium at the τ (42) scale of the mode crossing long before the transonic regime. kc = 2 1 2 . (1 Vˆ ) / − ||

As far as the dust drift is deeply subsonic, Vˆ|| 1, kc τ. This implies that slowly drifting dust tends to cause≪ the resonance≃ 5 RESONANCEOF MODES of three modes, since the frequency distance between two HSW 2ωs 0 as kc kJ . In contrast, the mode crossing shifts to high This Section is focused on the analytical approach to the new in- → → wavenumbers, k , as the drift velocity approaches the speed stabilities found in the vicinity of mode crossings, see the previous → ∞ of sound and the modes propagate downstream, Vˆ 1. In the Section. In the vicinity of the mode crossing defined by the condi- || → tion latter case the gas-dust dynamics should encounter the resonance of two modes, since the distance between the two HSW increases ωs = ωp ωc, (34) to high wavenumbers. The following Sections provide estimates of ≡ ∆ for the resonances of the both types. equation (28) can be written in a reduced form corresponding to the leading order in small f 1: ≪ 5.1 Resonance of three modes (ω ωs)(ω + ωs)(ω ωp)= ǫc, (35) − − This resonance is denoted as 3 hereafter. where the reduced form of the coupling term is Plugging equation (36) intoR equation (38) one obtains the fol- 2 2 lowing three roots: ǫc ǫ(k,ω = ωc)= if(τ iωc) . (36) ≡ − − 1 3 2 4 1 3 ∆ f / (ω + τ ) / Equation (35) is obtained by omitting the corrections f on ≈ c × ∼ 2 2 4π the LHS of equation (28), which become of the order higher than exp i arccos Ψ , exp i arccos Ψ i , (43) 3 3 ± 3 O(f) near ωc, and neglecting the change of the coupling term due      to the small deviation of ω from ωc. Equation (35) can be solved where accurately as a cubic equation with respect to ω, however, a further 2 ωc τ approximation can be done employing the smallness of the cou- Ψ − (44) ≡ √2(ω2 + τ 4)1/2 pling term, ǫc. c

Let the deviation from the mode crossing frequency caused by and ωc is defined by equation (41). It can be checked that the first the non-zero coupling term be ∆= ω ωc. As far as root in equation (43) gives the largest growth rate irrespective of − the values of τ and Vˆ||. It is instructive to consider the following limiting cases for ∆ 2ωs , (37) | | ≫ | | equation (43). ∆ is determined by equation i) The subsonic regime of strong self-gravity, Vˆ 1 and τ || ≪ ≫ 1/3 ˆ ∆ ǫ . (38) V||: ≈ c √3 i 1 3 4 3 In the opposite case, when ∆ i, ± − f / τ / . (45) ≈ 2   ∆ 2ωs , (39) | | ≪ | | ii) The subsonic regime of weak self-gravity, τ Vˆ 1: ≪ || ≪

√3 + i 1/3 2/3 2/3 4 ∆ i, ± f τ Vˆ . (46) which is restored after the mode coupling ≈ − 2 ||   MNRAS 000, 1–?? (2017) 10 V.V. Zhuravlev

Estimates of the growth rate given by equations (45) and (46) i) The subsonic regime of strong self-gravity, Vˆ|| 1 and τ ˆ ≪ ≫ match each other at V|| τ up to a factor of the order of unity. Vˆ||: Thus, the increase of the∼ drift velocity in the regime of weak self- f 1/2 τ 3/2 gravity leads to an additional increase in the growth rate as com- ∆ ( 1+i). (48) pared with the regime of strong self-gravity. The low power of ≈± 2 ˆ 1/2 − f V|| in both of these estimates explain the substantial growth rate of ˆ gas-dust perturbations in the medium, which would be marginally ii) The subsonic regime of weak self-gravity, τ V|| 1: ≪ ≪ stable at k & kJ in the absence of dust. In the regime of strong 1/2 f 1 2 1 2 self-gravity, which includes the case of no bulk drift of the dust, the / ˆ / ∆ τ V|| (1+i). (49) −1 ≈± 2 growth rate given by equation (45) measured in the units of tff be- 1/3 −1 ˆ haves like (fτ) . This implies that as τ exceeds unity, τ f , iii) The transonic regime of weak self-gravity, V|| 1 and τ → −1/2 → ≪ the dust drives clumping of matter at a rate approaching the char- δ , where δ 1 Vˆ 1: ≡ − || ≪ acteristic inverse free fall time. 1/2 1/2 ˆ f τ −1/4 Note that the transonic regime, V|| 1, in application to 3 (50) → R ∆ (2δ) (1 +i). is not considered in this study. As shown in Section 5.4, 3 is re- ≈± 2 ˆ R −1 stricted by the range V|| < 1 as long as τ does not exceed f , iv) The transonic regime of strong self-gravity, Vˆ|| 1 and τ 1 2 → ≫ which is unrealistic under the conditions in ISM. δ− / : Estimate (45) is in a good agreement with an accurate solution 1/2 3/2 f τ 1 4 of the general equation (28), see the bottom-left panel in Figure 2.It ∆ (2δ) / ( 1+i). (51) also reproduces the avoided crossing between the gas-dust modes ≈± 2 − seen in the top-left panel in Figure 2. Further, the more general Similarly to the case of 3, estimates of the growth rate fol- R estimate (43) remains good until the value of Vˆ corresponding lowing from equations (48) and (49) match each other at Vˆ τ || || ∼ to the change of the resonance type, see the bottom-right panel in up to the factor of the order of unity. The change of approximations Figure 2 and Section 5.4. As can be seen in the bottom-right panel from (i) to (ii) corresponds to the lowest growth rate, which is of in Figure 2, though equation (45) is still valid for parameters used the order of there, the more general equation (43) additionally reproduces the 1 2 ∆ i f / τ. (52) difference of the damping rates of gas-dust modes. ≃ The regime represented by equation (46) operates at rather The growth rate due to the resonance of HSW and SDW gener- ˆ small τ as well as V||, which are not presented in the numerical ally exceeds (52) for both Vˆ|| higher and lower than small τ. results considered above. However, it is included into analytical Thus, equation (52) can be used as a simple lower estimate of∼ the description of the general picture of modal resonances, see Section growth rate for 2. Note that estimate (52) expressed in the units −1 R 1/2 5.4. of tff yields simply if being independent of τ, i.e. the grain size. As far as τ is small, the regime of weak self-gravity continu- ously changes from subsonic (ii) to transonic (iii) variant with the 5.2 Resonance of two modes increase of Vˆ||. As can be seen from equations (49) and (50), this This resonance is denoted as 2 hereafter. leads to a further slow increase of the growth rate caused mainly by In this case, equation (36R) plugged into equation (40) yields the shift of the mode crossing to larger k. 1/2 2 2 For τ & 1, the value of the drift velocity corresponding to the f ωc τ + i(ωc + τ ) ∆ − , (47) lowest growth rate approaches unity, Vˆ 1. In this case, as Vˆ|| ≈± 2 1/2 → ωc increases from the small values, the subsonic regime of strong self- where ωc is again defined by equation (41). gravity (i) is replaced by the transonic regime of strong self-gravity As can be seen in Figures 2-5, equation (47) provides an ex- (iv), which leads to a further slow decrease of the growth rate, see −1/2 cellent estimate of the growth rate at the mode crossing located at equation (51). While Vˆ|| 1, δ becomes comparable to τ, (42). A poorer match is seen in the bottom-right panel in Figure which causes a continuous→ change to transonic regime of weak self- 2, where the influence of the third mode on the resonance of pos- gravity (iii). Again, the growth rate starts to slowly increase. Note itive frequency HSW and SDW is still considerable, see Section that the lowest growth rate corresponding to the change between 5.4. Equation (47) is also in accordance with the properties of the the regimes (iv) and (iii) is also estimated by equation (52). 2 frequencies of gas-dust modes. Indeed, as far as τ & ωc, what is As Vˆ goes back to zero, while τ and f remain constant, 2 || R shown in the top-right panel in Figure 2 and also in the top panel in must be replaced by 3, see the conditions (37) and (39). This can Figure 3, the gas-dust modes modified by a resonance undergo the occur in the regimes ofR either strong or weak self-gravity depending avoided crossing, and the growing mode passes below the damping on the value of τ, see Section 5.4 and tables 1, 2. mode, i.e. the shift of the growing/damping mode from the mode crossing is negative/positive, [∆] < 0 / [∆] > 0, in accordance ℜ 2 ℜ 5.3 Mode coupling with equation (47). In contrast, as τ . ωc, this situation is re- 2 versed, see Figure 5. The transitional case τ ωc is demonstrated Equation (47) indicates that once in Figure 4, when the modes undergo coupling,≃ i.e. they coalesce 2 with each other giving birth to the coupled modes. The coupled ωc = τ , (53) modes are represented by a complex conjugate pair of . Their fre- ω ∆ becomes imaginary. Thus, the gas-dust modes akin to the posi- quencies are identical to each other in accordance with equation tive frequency HSW and SDW exhibit identical frequencies5 equal (47). The special case of the mode coupling is considered in a more detail in Section 5.3. Equation (47) is more tractable in certain limiting cases. 5 and also phase velocities

MNRAS 000, 1–?? (2017) Gravitational instability of gas-dust mixture 11

Table 1. Map of the analytical limiting cases for the resonance of modes. The parameter τ increases from the left to the right column, while it is assumed that τ ≪ 1. It is implied that τ remains constant for each column, ˆ while V|| increases from top to the bottom of the columns. The upper index after R2,3 denotes the number of the limiting case collected in Sections 5.1 and 5.2. 0,1 f < τ ≪ τ ′ τ ∼ τ ′ τ ′ ≪ τ

ˆ ≪ Ri ˆ ≪ Ri ˆ ≪ ˆ ′ Ri 0 6 V|| τ : 3 0 6 V|| τ : 3 0 6 V|| V|| : 3 τ ≪ Vˆ ≪ Vˆ ′′ Rii τ ≪ Vˆ ≪ Rii Vˆ ′ ≪ Vˆ ≪ τ Ri ^ || || : 3 || 1: 2 || || : 2 Vˆ ′′ ≪ Vˆ ≪ Rii ˆ → Riii ≪ ˆ ≪ Rii | || || 1: 2 V|| 1: 2 τ V|| 1: 2 ˆ iii ˆ iii V|| → 1: R2 ———– V|| → 1: R2

0,01

Table 2. The same as in the Table 1 for τ & 1. 0,01 0,1

τ ∼ 1 1 ≪ τ

Vˆ ≪ Vˆ ′ Ri Vˆ ≪ Vˆ ′ Ri 0 6 || || : 3 0 6 || || : 3 ˆ ′ ≪ ˆ ≪ Ri ˆ ′ ≪ ˆ ≪ Ri V|| V|| 1: 2 V|| V|| 1: 2 Figure 6. Graphic representation of the limiting cases for the resonance of ˆ iii ˆ iv Vˆ ≪ f . V|| → 1: R2 V|| → 1: R2 modes for || 1. The solid line is given by equation (59) for = 0 01, ˆ iii ˆ ———— V|| → 1: R2 while the dashed line shows V|| = τ.

5.4 Changing of resonance type to those HSW and SDW themselves have at the mode crossing. At The restrictions on the existence of 3 and 2, see equations (37) the same time, damping and growth rates of modes have the same and (39), lead to the condition for theR changeR of the resonance type absolute value 2ωc ∆ , (56) 1 2 ∼ | | ∆ if / τ, (54) ≈± which yields the critical value of the dust fraction 3 which recovers an order-of-magnitude lower estimate of the growth ′ 8 ωc f = 2 2 . (57) rate in 2, see equation (52). In (single-)fluid dynamics, such ωc + τ R modes are referred to as the coupled modes after Cairns (1979) It is instructive to rewrite equation (57) in terms of the drift ve- who applied the concept of mode coupling to the explanation of the locity. One obtains in the leading order over the small dust fraction, Kelvin-Helmholtz instability. The condition of the mode coupling f 1: (53) can be expressed with respect to drift velocity ≪ ′ 1 1/3 4/3 f τ ωc f τ + , (58) ˆ (55) ≈ 2 24 V|| = 2 1/2 , (1 + τ ) ′ where ωc denotes the critical value of the mode crossing frequency corresponding to transition between 2 and 3 for specified f and which gives ˆ for the case shown in Figure 4. The R R V 0.7 τ = 1.0 τ. The corresponding critical drift velocity reads accurate solution≈ shown in Figure 4 demonstrates that the coupled 1 1 3 1 3 f modes exist in some interval around the mode crossing producing Vˆ cr f / τ / + (59) a distinctive “bridge” of instability (Glatzel 1988). The simplified || ≈ 2 24τ model of coupling between HSW and SDW, which leads to the re- provided that duced dispersion equation (35) with the real coupling term, makes 1 f − τ & f. (60) it possible to quantify the energy of modes involved in resonance. ≫ As expected for problems of this kind, HSW and SDW coalescing The latter condition guaranteers that (59) remains small compared into the coupled modes have positive and negative energies, respec- to unity. Otherwise, equation (59) is not valid anymore. As soon as cr tively, see the Appendix B for details. According to the common Vˆ Vˆ , the modes come into 3, whereas in the other case || ≪ || R interpretation of the corresponding instability in such a problems, they come into 2. Equation (59) contains two terms, which are the growth of perturbations is caused by the energy flow from SDW denoted as R having negative energy to HSW having positive energy. The total 1 1 3 1 3 energy of the system of modes remains unchanged. However, the Vˆ ′ f / τ / (61) || 2 amplitude of the negative energy mode losing the energy grows ex- ≡ ponentially. Conversely, the amplitude of the positive energy mode and grows because it receives energy. A related example of the mode f Vˆ ′′ . (62) coupling in the two-fluid dynamics of perturbations has been stud- || ≡ 24τ ied recently by Zhuravlev (2019), who showed that it takes place As long as τ τ ′, where on the background of the dust settling to the midplane of a proto- ∼ 1 2 planetary disc. τ ′ (f/8) / , (63) ≡ MNRAS 000, 1–?? (2017) 12 V.V. Zhuravlev

Vˆ ′ Vˆ ′′ τ. In the limit of small τ τ ′ there is an inequality At the same time, equation (46) is valid as far as Vˆ Vˆ ′′ 1, || ∼ || ∼ ≪ || ≪ || ≪ Vˆ ′′ Vˆ ′ τ, whereas in the limit of large τ τ ′, conversely, which reads || ≫ || ≫ ≫ ˆ ′′ ˆ ′ V|| V|| τ. Thus, the drift velocity corresponding to the tran- f ≪ ≪ kc (69) ˆ ′ ˆ ′′ ≪ ˆ sition 2 3 approximately equals V|| and V|| in the case when 24V|| R ↔ R ′ τ is, respectively, smaller and larger that τ . Consequently, as the ′ after making use of equation (62). Thus, the weak self-gravity of dust drift intensifies in the case τ τ , the transition from 3 to ≫ R gas-dust medium with the subsonic bulk drift of the dust extends 2 occurs in the strong self-gravity regime, Vˆ τ, at the drift R || ≪ the resonance of three modes and the corresponding value of the velocity given by equation (61). The corresponding approximate ′ growth rate onto much shorter length-scales up to kc τ , see expressions for ∆ given by equations (45) and (48) continuously equation (63). ∼ ˆ ˆ ′ replace one another at V|| = V||. On the other hand, as the dust drift ˆ ′ As soon as V|| 1, equation (64) reduces to intensifies in the case τ τ , the transition from 3 to 2 occurs → ≪ R R in the weak self-gravity regime, Vˆ τ, at the critical drift veloc- 2 k || ∆ = if (70) ity given by equation (62). Again, the≫ corresponding approximate 2 expressions for ∆ given by equations (46) and (49) continuously provided that ˆ ˆ ′′ replace one another at V|| = V|| (1 Vˆ )k ∆ k. (71) For a clear exposition of the analytical results, the overall pic- − || ≪ ≪ ˆ ture of modal resonances for any τ and V|| is shown in the Tables The solution of equation (70) reads 1, 2 and graphically in Figure 6. 1 + i 1 2 1 2 It is worth comparing the lower estimate of the growth rate ∆= f / k / , (72) in 2, see equation (52) (or equation (54)), with the approximate ± 2 R growth rate in 3 in the absence of the dust drift, Vˆ = 0, see 1/2 R || which recovers the dependence k of the supersonic acous- equation (45). They are comparable to each other provided that τ tic RDI at intermediate wavelengths∝ shown in Figure 1 of HS18. 1/2 ′ ′ ∼ f , which is of the order of τ . Hence, in the limit of τ τ 3 Therefore, the intermediate-wavelength HS18 instability in the ′ ≫ R taking place at Vˆ Vˆ provides a higher growth rate than 2 ≪ || R transonic regime can be considered as 2 in the non-self- taking place, at least, for Vˆ τ Vˆ ′ , i.e. sufficiently close to gravitating medium. Since there is no mode crossingR in the absence ∼ ≫ || the mode coupling, see Section 5.3. This trend is increasing as τ of self-gravity, the sound wave (SW hereafter) and SDW propagat- approaches unity or becomes higher than unity. ing in the same direction can come into resonance provided that the dust fraction is sufficiently high, or conversely, the drift velocity is sufficiently close to the sonic value, which is expressed by the LHS 5.5 Connection with HS18 instability of inequality (71). The solution (72) is also identical to equation 1/2 (50) after the replacement k kc τ/(2δ) . Note that be- Let the general dispersion equation (28) be considered in the ab- cause this solution is produced→ by the≈ mode crossing of HSW and sence of self-gravity, τ = 0, and, additionally, in the limit of the SDW, it is valid in the regime of weak self-gravity for any non-zero small dust fraction. Once ∆ is a small deviation from an exact so- dust fraction. By the same reason, the counterpart of this solution ˆ lution of equation (28) taken in the limit f 0 as ω = V||k, it in the subsonic regime given by equation (49) is also valid for any → obeys the following equation small dust fraction as well as for any small Vˆ max τ, Vˆ ′′ : || ≫ { || } 2 2 see the tables 1 and 2. (Vˆ k + k + ∆)(Vˆ k k +∆)∆= ifVˆ k (64) || || − || An explicit form of the restriction (71) after it is combined derived in the leading order in f. The restriction with the solution (72) reads k f (1 Vˆ )k. (73) ∆ k (65) ≫ ≫ − || ≫ The RHS of this inequality implies that 2 in the non-self- yields the reduced equation for ∆ gravitating medium recovers the growth rateR attained in the cor- ii iii 3 ˆ 2 2 responding regime of weak self-gravity, i.e. both 2 and 2 , see ∆ = ifV|| k , (66) R R the tables 1 and 2, for Vˆ|| very close to sonic value only. The lat- and the corresponding solution ter explains the 2 bumps on the profiles of the growth rate for R τ = 0.01, 0.1, 1.0 substantially exceeding the profile of HS18 √3 + i 1 3 2 3 2 3 ∆= i, ± f / k / Vˆ / . (67) instability in panels (c-f) in Figure 1. In other words, while the − 2 ||   drift velocity gradually increases up to the sonic value, the crossing It is seen that equation (67) recovers equation (18) of HS18. There- modes approach each other farther and farther from k = kc caus- ing the expansion of the zone of resonance around k = k , which is fore, the long-wavelength HS18 instability can be considered as 3 c in the non-self-gravitating medium. The solution (67) is also iden-R seen as the growth rate of HS18 instability approaching the bumps from below. tical to equation (46) after the replacement k τ. This is because → τ standing in equation (46) originates from kc τ > 0 in the ≈ limit of the small drift velocity. However, since ωs(kc) 0 in the ≈ self-gravitating medium, the corresponding restriction on the oc- 6 ASTROPHYSICAL IMPLICATIONS currence of 3 is much weaker for the subsonic drift. Indeed, the validity of equationR (66) follows from the restriction (65) combined The Epstein aerodynamic drag yields the stopping time with the solution (67). Explicitly, (Weidenschilling 1977): ρms ˆ 2 ts = , (74) k fV|| . (68) ρ v ≪ g th MNRAS 000, 1–?? (2017) Gravitational instability of gas-dust mixture 13

where s is the grain size and the mean thermal velocity vth = the other hand, as Vˆ|| 1 the growth rate for τ = 0.25 behaves as 1/2 −1/4 → (8/π) cs. 0.1(2δ) , where δ 1 Vˆ||, see equation (50). Estimate of τ with the help of equation (74) reads ∼ Thereby, in all cases≡ discussed− just above, the growth rate of −1 the new instability is a considerable fraction of tff . Also, it weakly depends on the drift velocity covering the whole subsonic band 1 2 − / −1/2 s ρm T n from 0 to 1. As the size of the cloud passes the Jeans length in τ 0.25 , ≃ 10−4cm 3gcm−3 50K 50 cm−3 the course of its formation, its gravitational contraction proceeds        (75) on the longer time-scale as compared with tff . The resonant insta- where the temperature and gas density are normalised by plausible bility due to either 2 or 3 may have enough time to launch the average values for the young molecular cloud a few Myr old, when collapse of the sub-JeansR massR cloud. How much the new instabil- its Jeans mass falls below its own mass, see e.g. numerical simu- ity affects the gravitational collapse of the cloud, will depend on its lation by V´azquez-Semadeni et al. (2007). In the course of its evo- cooling rate. lution, a molecular cloud fragments into denser and cooler struc- It is important to note that the new instabilities provide the tures. It finally gives birth to prestellar cores with typical density growth of the dust fraction. Yet, it should be emphasised that the 5 3 and temperature n 10 cm− and T 10K, respectively, see gas is also considerably affected by the instability in spite of the ≃ ≃ e.g. numerical simulation by Masunaga & Inutsuka (2000). There- small value of the background dust fraction. For modes, this means fore, in prestellar cores τ 0.01 indicates that the micron-sized that the amplitude of the relative perturbation of the gas density is ≃ grains are stronger coupled to the gas as the dynamical timescale not negligible in comparison with that of the relative perturbation is defined by the self-gravity of the gas-dust mixture. The range of the dust fraction. The latter can be checked using equation(21), 0.01 . τ . 0.25 may shift both up and down depending on the which is taken for the growing modes at 3 and 2. It is found that R ′ R actual grain size. According to the conventional dust model in the the ratio of the Fourier amplitudes δ¯ and h¯ is finite provided that diffuse ISM, see Draine (2003), the peak of the grain size distri- both f and τ (or Vˆ||) are finite. For example, in the case Vˆ|| = 0 bution is attained at sub-micron scales, s 0.3µm. The size of and τ 1 it reads 3 ≃ grains in clouds with n > 100 cm− is less clear, as they are ≪ τ 2/3 able to accrete volatile elements increasing both s and τ and conse- ¯ ¯′ (76) δ 1/3 h . quently the mass fraction accumulated in dust (K¨ohler et al. 2015). ≈ f Moreover, in the dense regions of molecular clouds (cores), where In the limit f 0 the inequality δ¯ h¯′ is always satisfied, so that n 104 cm−3 and higher, grains coagulate with each other attain- → ≫ ∼ the instability of the gas-dust mixture is provided mostly by the rel- ing sizes as large as s 10µm, see e.g. Ormel et al. (2009). This ′ ∼ ative perturbation of the dust density, ρp/ρp, while that quantity for is confirmed by the observational evidence for micron-sized grains the gas is negligible. However, the values of f and τ present in the in molecular clouds, see e.g. Pagani et al. (2010), Saajasto et al. ISM, as well as their low powers entering equation (76), make δ (2018), as well as by the indication of the dust fraction variations and h′ comparable to each other, especially in dense clouds. As the within a particular molecular cloud (Liseau et al. 2015). The fol- Jeans scale becomes much smaller than the size of the cloud, the lowing conclusions about the growth rate of the new resonant in- gravitational contraction enters the free fall stage, when the equiva- −3 stabilities are made for nominal ρm = 3gcm , s = 1µm and lence principle of gravity freezes the grains into common free mo- ˆ f = 0.01 though various τ and V|| with the help of the tables 1 and tion of matter, i.e. the growth of the dust fraction is stalled, see the 2. comments in the Appendix A1. How strong the concentration of As long as τ . τ ′ 0.035 for f = 0.01, see equation dust gets during the time passed from the sub-Jeans contraction of ≃ (63), the lower estimate of the growth rate (given in units of the the cloud up to its free fall is an issue for future research. As soon as 1 3 inverse free fall time in this Section) is 0.22τ / attained in the the bulk drift of the dust is significant, the growing dust overdensi- i ∼ ˆ regime 3 for the sufficiently slow dust drift V|| . τ . This value ties are carried by the unstable gas-dust wave which is akin to SDW. R weakly depends on the grain size and yields the lowest value of During the non-linear stage occurring in a sufficiently dense cloud, the growth rate 0.05 for τ = 0.01. Further, the largest growth these dust overdensities may become opaque to ambient radiation ∼ ii ˆ rate attained in the regime 3 for V|| 0.04 corresponding to forcing the dust drift. At this moment, the dust drift is suppressed R ≃ the change of the resonance type in this case (see equation (62) ) and the dust overdensities lag behind the wave front, which means ˆ is equal to 0.12. For a higher drift velocity, e.g. V|| = 0.1, one that the resonance ceases to operate. The gas-dust wave becomes an ∼ ii should use the estimate of the growth rate for the regime 2 ob- ordinary HSW running away from dusty domains. The latter effect R taining 0.32, which is almost two orders of magnitude higher additionally increases the final dust fraction as it removes dense gas ∼ than the corresponding growth rate of the intermediate wavelength out of the domains with the enhanced dust concentration. This is an HS18 instability, see panel (b) in Figure 1. As discussed in Section other way that dusty domains may be produced at the sub-Jeans 5.5, equations (46) and (49) recover the corresponding results of scales of dense clouds. HS18 obtained for the non-self-gravitating medium. However, they The dusty prestellar cores, which may form within the sug- are valid for the deeply subsonic dust drift, since SDW falls in res- gested scenarios, may potentially give birth to big young protoplan- onance with HSW, which is (are) slowed down by the self-gravity etary discs (see Lee et al. (2017a) and Lee et al. (2018) for recent of gas. As a less coupled dust and gas are considered, τ & 0.035 observations of such systems). Though the majority of these Class for f = 0.01, the lower estimate of the growth rate is simply the 0 discs have smaller sizes, see Ansdell et al. (2018), the formation square root of the dust fraction, 0.1, given by equation (52). of bigger discs is not exceptional. Moreover, it is challenging from ˆ ∼ This value is attained at V|| τ and independent of the grain size. a theoretical view, see Hennebelle et al. (2016) and the review by ∼ For example, as Vˆ|| = 0.5, the growth rate is an order of magni- Zhao et al. (2020). Numerical simulations of the magnetised core tude higher than that of intermediate-wavelength HS18 instability, collapse have shown that the magnetic braking should prevent the see panel (c) in Figure 1. If there is no dust drift, Vˆ 0, the young disc formation when dissipation of the magnetic flux is ne- || → growth rate becomes 0.14 for a particular value τ = 0.25. On glected, see Mellon & Li (2008). Considering the dissipative ef- ∼ MNRAS 000, 1–?? (2017) 14 V.V. Zhuravlev

6 fects corrects the situation (see e.g. Masson et al. (2016) ). How- 3 and 2, see equations (38) and (40), respectively. As one fol- ever, it reveals that Class 0 disc size is defined mostly by the am- lowsR theR origin of the coupling term from the general equations bipolar diffusion of magnetic field, which in turn considerably de- (20-22), it becomes clear that the coupling term emerges from the pends on the distribution of the grains’ size (see Zhao et al. (2016), gravitational dust back-reaction on gas in the regime of strong self- Dzyurkevich et al. (2017) and Zhao et al. (2018)). The amount of gravity. The latter is introduced by the additional small gravita- very small charged grains may be reduced by the dust coagu- tional attraction provided by an excess of the dust in gas-dust mix- lation, which occurs more efficiently in the dusty environment ture. In turn, an excess of dust is caused by the relative velocity of (Ormel et al. 2009). dust to gas in the perturbed dynamics. Therefore, in the regime of strong self-gravity (including the case Vˆ|| = 0), dust back-reaction on the gas has nothing to do with the bulk drift of the dust. On the other hand, the coupling term emerges from the aerodynamical dust 7 SUMMARY back-reaction on gas in the regime of weak self-gravity. The latter is introduced by perturbations of the density of dust moving at the A partially coupled gas-dust mixture is subject to GI at scales velocity of the bulk drift, which is similar to the case of the HS18 smaller than the Jeans length-scale. An unbounded uniform instability. Moreover, the corresponding estimates of the growth medium becomes significantly unstable for a small fraction of dust, rates in 3 and 2 recover that of HS18 after the replacement which can be explained by the resonant nature of the new instabil- R R k k given by equation (42). Indeed, the long- and intermediate- ity. The scale of resonance is defined by the crossing of gas modes c wavelength→ HS18 instability may be considered as, respectively, (two oppositely propagating HSW) with the dust mode (SDW) 3 and 2 in the non-self-gravitating medium, see Section 5.5. existing in a mixture in the limit of the negligible dust fraction, However,R R it becomes much stronger on account of self-gravity due f 0. As the dust is suspended in gas, SDW is formally a static → to the mode crossing of HSW and SDW at kc >kJ . In contrast, the wave coming to resonance with both HSW. This resonance denoted 7 ordinary SW always have higher phase velocity than SDW requir- here as 3 occurs strictly at the Jeans scale, kc = kJ , see Section R ing the much higher dust fractions for these mode to come into res- 5.1. The growth rate of GI corresponding to 3 evaluated as the onance. Additional remarks on the physics of the long-wavelength solution of the reduced dispersion equation (35R) is determined by HS18 are given in Appendix C. f 1/3. As there is subsonic bulk drift of the dust, 0 < Vˆ < 1, SDW Generally, the coupling term (36) takes a complex value, propagates with phase velocity equal to Vˆ projected onto wavevec- which causes the avoided crossing of modes akin to HSW and SDW tor denoted as Vˆ . In this case, it comes into resonance with HSW || for f > 0, see Section 4.2 and the top-right panel in Figure 2 as propagating in the same direction with equal phase velocity. This well as the top panels in Figures 3 and 5. However, for the particu- resonance denoted here as 2 takes place at k > k , see Section c J lar value of the drift velocity given by equation (55), the coupling 5.2. According to equationR (35), the growth rate of GI correspond- 1/2 term (36) becomes real, which means that as f > 0, the crossing ing to 2 is determined by f . HSW and SDW give birth to the coupled modes represented by a AnR important free parameter of the considered model is the complex conjugate pair of solutions of the reduced dispersion equa- ratio of the stopping time of the grains to the free fall time, τ, tion, see Section 5.3. It is not difficult to gain further insight into the which quantifies the relative strength of the dust to gas dynami- physics of GI, which appears as bumps on curves in the bottom pan- cal coupling and the mixture self-gravity. The analytical approach els in Figure 4 in this case. The energy of modes akin to HSW and to the problem shows that simple estimates of the growth rate can SDW outside the band of the mode coupling can be obtained as- be obtained in the regimes of weak self-gravity, τ Vˆ 1, and ≪ ≪ suming that they are approximately neutral, see the Appendix B for strong self-gravity, Vˆ τ along with Vˆ 1, see equations || ≪ || ≪ the description of the corresponding simplified model. The modes (45-46) and (48-49) for the resonances 3 and 2, respectively. coalescing inside the band of GI have the energies of the opposite These estimates may be useful in variousR applications.R The general signs. Thus, the conserved total energy flows from the negative en- picture of all limiting cases is represented in tables 1 and 2 and, ergy mode (SDW) to the positive energy mode (HSW) causing the additionally, in Figure 6. For particular values of τ and f, there is growth of their amplitudes. a critical value of the drift velocity, Vˆ cr, which corresponds to the || GI of the gas loaded with a suspended dust which is pro- boundary between 3 and 2, see equation (59). The critical drift ˆ R R cr ′ duced by 3 for strictly V|| = 0 can be considered as the missing velocity takes the minimum Vˆ τ at τ τ given by equation || link in theR description of the dynamics of a partially coupled self- (63). The analytical results show∼ that for a∼ particular, sufficiently gravitating gas-dust mixture. This is the link between the limiting small τ . τ ′, the lower estimate of the growth rate as function 1 3 1 3 1 case of the free falling dust-laden gas corresponding to k 0, of the subsonic drift velocity equals to f / τ / t− . This value is ff see the Appendix A1, and the limiting case of the dust settling→ ˆ ′ attained for V|| . τ. Otherwise, as τ & τ , such a lower estimate through the gas being in hydrostatic equilibrium in its own grav- 1/2 −1 ˆ equals to f t , which is attained at V|| τ. It is discussed that itational well corresponding to k , see the Appendix A2. This ff ∼ in molecular clouds the growth rate of GI of the gas-dust mixture is illustrated by the dot-dashed curve→∞ of the GI growth rate in the at sub-Jeans scales can attain significant fraction of the inverse free bottom-left panel in Figure 2. fall time, see Section 6. In future, the dynamic role of dust in the GI of dense ISM The growth rate of GI in the vicinity of resonance is deter- should be studied in the framework of the global linear stabil- mined by the coupling term given by equation (31). For small ity analysis of real configurations. This will help to see, whether f 1, the coupling term reduces to equation (36) at the mode the self-gravitating gas-dust mixture with a small fraction of dust ≪ crossing, which is used to obtain the approximate growth rates for can be most unstable with respect to perturbations growing due to

6 For the most recent studies see also Machida & Basu (2019) and Lam et al. (2019) 7 Technically, in this case the mode crossing is located at k = 0

MNRAS 000, 1–?? (2017) Gravitational instability of gas-dust mixture 15

3 or 2. Of course, accompanying numerical simulations are re- Binney J., Tremaine S., 1987, Galactic dynamics quired.R R Bonnor W. B., 1956, MNRAS, 116, 351 GI of a mixture with suspended dust, Vˆ|| = 0, considered at Cairns R. A., 1979, Journal of Fluid Mechanics, 92, 1 the Jeans scale induces the growing relative velocity of dust and Chandrasekhar S., Fermi E., 1953, ApJ, 118, 116 gas, v. As the grain size is distributed in some range, the grow- Draine B. T., 2003, ARA&A, 41, 241 ing relative velocity of grains of different size should trigger the Draine B. T., 2011, Physics of the Interstellar and Intergalactic Medium enhanced growth of grains via coagulation. In turn, the growth of Dzyurkevich N., Commerc¸on B., Lesaffre P., Semenov D., 2017, A&A, 603, A105 grains should cause the increase of their stopping time, t , and con- s Elmegreen B. G., Elmegreen D. M., 1978, ApJ, 220, 1051 sequently, the further decoupling of the gas-dust mixture along with Fabrikant A. L., Stepanyants Y. A., Stepaniants I. A., 1998, Propagation of the following speedup of its gravitational contraction. Therefore, Waves in Shear Flows. World Scientific Pub Co Inc the model for dynamics of self-gravitating gas-dust mixture should Fiege J. D., Pudritz R. E., 2000, MNRAS, 311, 85 be extended to account for grains’ growth. Flannery B. P., Krook M., 1978, ApJ, 223, 447 On the other hand, turbulence generally weakens dynamic in- Friedman J. L., Schutz B. F., 1978, ApJ, 221, 937 stability. The ISM is also turbulent. Its role in damping of the Girichidis P., et al., 2020, arXiv e-prints, p. arXiv:2005.06472 growth rates obtained in this study is still to be determined. Glatzel W., 1988, MNRAS, 231, 795 The case τ > 1 describes weakly coupled dust and gas. While Hennebelle P., Commerc¸on B., Chabrier G., Marchand P., 2016, ApJ, not relevant for typical conditions in ISM (see Section 6), it, how- 830, L8 ever, may be appropriate in the densest prestellar cores as nurseries Hopkins P. F., Lee H., 2016, MNRAS, 456, 4174 of the big particles 1mm born due to effective coagulation of Hopkins P. F., Squire J., 2018a, MNRAS, 479, 4681 typical micron-sized∼ grains. The weak coupling of big particles to Hopkins P. F., Squire J., 2018b, MNRAS, 480, 2813 Inutsuka S.-i., Miyama S. M., 1997, ApJ, 480, 681 the gas invalidates the description of dust as a pressureless fluid. Jacquet E., Balbus S., Latter H., 2011, MNRAS, 415, 3591 For consistency, one should additionally account for the non-zero Jeans J. H., 1902, Philosophical Transactions of the Royal Society of London Series A, velocity dispersion of the dust in this case. 199, 1 The model of gas-dust perturbations considered here does not Johansen A., Blum J., Tanaka H., Ormel C., Bizzarro M., Rickman H., take into account perturbation of ts due to perturbation of gas den- 2014, Protostars and Planets VI, pp 547–570 sity. It can be checked that this extension of the model does not K¨ohler M., Ysard N., Jones A. P., 2015, A&A, 579, A15 affect the results obtained for 2,3. However, it may be impor- Krause M. G. H., et al., 2020, arXiv e-prints, p. arXiv:2005.00801 tant far from the mode crossingR along with the other non-resonant Lam K. H., Li Z.-Y., Chen C.-Y., Tomida K., Zhao B., 2019, MNRAS, corrections contained in the general dispersion equation (28). The 489, 5326 non-resonant contribution to GI of the gas-dust mixture may be im- Larson R. B., 1969, MNRAS, 145, 271 portant as the dust fraction is not small. This issue can be addressed Larson R. B., 1985, MNRAS, 214, 379 in a future work. Latter H. N., Rosca R., 2017, MNRAS, 464, 1923 Ledoux P., 1951, Annales d’Astrophysique, 14, 438 At last, the charged grains are affected by Coulomb drag and Lee C.-F., Li Z.-Y., Ho P. T. P., Hirano N., Zhang Q., Shang H., 2017a, Lorentz force if mixed with ionised and magnetised gas. GI of such Science Advances, 3, e1602935 a mixture consisting of the charged dust and weakly ionised plasma Lee H., Hopkins P. F., Squire J., 2017b, MNRAS, 469, 3532 is another problem to be resolved. Lee C.-F., Li Z.-Y., Hirano N., Shang H., Ho P. T. P., Zhang Q., 2018, ApJ, 863, 94 Liseau R., et al., 2015, A&A, 578, A131 DATA AVAILABILITY Machida M. N., Basu S., 2019, ApJ, 876, 149 Masson J., Chabrier G., Hennebelle P., Vaytet N., Commerc¸on B., 2016, No new data were generated or analysed in support of this research. A&A, 587, A32 Masunaga H., Inutsuka S.-i., 2000, ApJ, 531, 350 Mattsson L., Bhatnagar A., Gent F. A., Villarroel B., 2019, MNRAS, ACKNOWLEDGMENTS 483, 5623 Mellon R. R., Li Z.-Y., 2008, ApJ, 681, 1356 I thank Natalia Dzyurkevich for discussing the formation of the Miyama S. M., Narita S., Hayashi C., 1987, Class 0 discs. I am grateful to Artem Tuntsov and Henrik Latter for Progress of Theoretical Physics, 78, 1273 careful reading of the manuscript and their useful comments and Monceau-Baroux R., Keppens R., 2017, A&A, 600, A134 suggestions that helped to improve the presentation of the study. Moseley E. R., Squire J., Hopkins P. F., 2019, MNRAS, 489, 325 Of course, this work would not have been done if it were not for Nagasawa M., 1987, Progress of Theoretical Physics, 77, 635 the dedication of my family. I acknowledge the support from the Naranjo-Romero R., V´azquez-Semadeni E., Loughnane R. M., 2015, ApJ, 814, 48 Foundation for the Advancement of Theoretical Physics and Math- Ormel C. W., Paszun D., Dominik C., Tielens A. G. G. M., 2009, A&A, ematics “BASIS”. Additionally, this work was supported in part 502, 845 by the Government and the Ministry of Science and Higher Ed- Pagani L., Steinacker J., Bacmann A., Stutz A., Henning T., 2010, Science, ucation of the Russian Federation (project no. 075-15-2020-780) 329, 1622 and in part by the Program of development of Lomonosov Moscow Penston M. V., 1969, MNRAS, 144, 425 State University. Saajasto M., Juvela M., Malinen J., 2018, A&A, 614, A95 Squire J., Hopkins P. F., 2018a, MNRAS, 477, 5011 Squire J., Hopkins P. F., 2018b, ApJ, 856, L15 References Takahashi S. Z., Inutsuka S.-i., 2014, ApJ, 794, 55 Tricco T. S., Price D. J., Laibe G., 2017, MNRAS, 471, L52 Ansdell M., et al., 2018, ApJ, 859, 21 V´azquez-Semadeni E., G´omez G. C., Jappsen A. K., Ballesteros-Paredes J., Bate M. R., Lor´en-Aguilar P., 2017, MNRAS, 465, 1089 Gonz´alez R. F., Klessen R. S., 2007, ApJ, 657, 870

MNRAS 000, 1–?? (2017) 16 V.V. Zhuravlev

V´azquez-Semadeni E., Palau A., Ballesteros-Paredes J., G´omez G. C., which describes a slow collapse of self-gravitating dust drifting Zamora-Avil´es M., 2019, MNRAS, 490, 3061 through the gas in hydrostatic equilibrium. Such a collapse is not a Weidenschilling S. J., 1977, MNRAS, 180, 57 free fall: it is restrained by the aerodynamic drag. The dimension- Weingartner J. C., Draine B. T., 2001, ApJ, 553, 581 less growth rate of this kind of dust clumping reads Whipple F. L., 1972, in Elvius A., ed., From Plasma to Planet. p. 211 2 Whitham G., 2011, Linear and Nonlinear Waves. Pure and Applied Mathe- ω ifτ . (A5) matics: A Wiley Series of Texts, Monographs and Tracts, Wiley → Youdin A. N., 2005, arXiv e-prints, pp astro–ph/0508659 It can be checked that equation (A5) is in agreement with an accu- Youdin A. N., 2011, ApJ, 731, 99 rate curves plotted at the panel (a) in Figure 1. Youdin A. N., Goodman J., 2005, ApJ, 620, 459 Zhao B., Caselli P., Li Z.-Y., Krasnopolsky R., Shang H., Nakamura F., 2016, MNRAS, 460, 2050 A3 Jeans wavelength Zhao B., Caselli P., Li Z.-Y., 2018, MNRAS, 478, 2723 Zhao B., et al., 2020, Space Sci. Rev., 216, 43 In the limit of the negligible dust fraction, f 0, an additional Zhuravlev V. V., 2019, MNRAS, 489, 3850 simplifying assumption →

tev tff (A6) ≫ APPENDIX A: TOWARDS INTERPRETATION OF GI OF is justified at the considered scale k kJ , since one deals with DUST-LADEN MEDIUM either a slowly propagating sound wave≃ or weak Jeans instability, It is assumed here that dust with small fraction, f 1, is sus- when ≪ pended in gas, Vˆ|| = 0. The dynamics of small perturbations of ′ h gg (A7) self-gravitating homogeneous medium is considered in the three ∇ ≈ basic cases, k 0, k kJ and k . In all these cases, the with gg being the gravitational acceleration arising from the gas → ≈ → ∞ analysis starts from equations (20-22) and it is assumed for sim- self-gravity. Since for small f 1 the new solution discussed plicity that here slightly differs from this basic≪ case, it is reasonable to use the restriction (A6). The assumptions (A1) and (A6) make the inertial tev ts, (A1) ≫ terms on the LHS of equation (22) small compared with the leading where tev is the characteristic time of evolution of a mixture. terms on the RHS of equation (22), i.e. the terms remaining there in the limit f 0. Thus, there is an approximate balance →

A1 Long-wavelength limit v τωff h1. (A8) ∇ · ≈− As k 0, the perturbation of the pressure gradient becomes neg- Equation (A8) resembles the balance of terms in the terminal veloc- → ligible and the gas undergoes free fall uniformly with the dust. ity approximation applicable for certain problems of gas-dust dy- Clearly, there is no perturbation of the relative velocity of gas and namics in protoplanetary discs, see e.g. Youdin & Goodman (2005) dust, which is demonstrated by equation (22), where all terms ex- and Zhuravlev (2019). Further, equation (A8) implies that the term cept those containing v cancel each other by means of equation v on the RHS of equation (20) along with the addition f ∇ · ∼ ∇ · ′ ∼ (20). As v 0 drives the relative perturbation of the dust in front of h in the square brackets on the RHS of this equation ∇ · → fraction, δ, see equation (21), it is clear that can be omitted as they are of the order of a small correction to the single-fluid GI growth rate due to a slight increase of the total den- δ 0 (A2) ≈ sity with the account of the dust. Also, equation (A8) is used to in the course of collapse. In this way, one obtains the growth rate reduce equation (21). of gas-dust density approaching Finally, the reduced set of equations can be expressed as

f ∂ug ω iτ 1+ (A3) = gp, (A9) → 2 ∂t   in units of ts. This equation recovers the inverse free fall time gen- erally expected for the gas-dust mixture with grains frozen in fluid 1 ∂h′ 2 = ug, (A10) elements of gas. cs ∂t −∇ · This may be the reason for a weak segregation of (sub-

)micron-sized grains with gas found in the numerical simulations ′ of collapse of gravitationally unstable Bonnor-Ebert sphere laden ∂δ h = τωff 2 , (A11) with partially coupled dust, see Bate & Lor´en-Aguilar (2017). ∂t cs

2 A2 Short-wavelength limit gp = fωff δ, (A12) ∇ · −

As the wavenumber becomes high, k , while tev is kept con- where gp Φp is the gravitational acceleration caused by →∞ ≡ −∇ stant, the amplitude of h1 vanishes. This reproduces the incom- deviation of the dust fraction from its background value. It is gp pressible gas with the velocity free of divergence, ug 0. standing on the RHS of equation (A9) that represents the dust grav- ∇ · → Thus, equation (22) yields v fτωff δ and equation (21) itational back-reaction on gas making the coupling term entering takes the form ∇ · → − the dispersion equation (28) non-zero in the absence of the bulk ∂δ drift of the dust, see the description of the corresponding resonant fτ ωff δ, (A4) ∂t ≈ instability in Section 5.1.

MNRAS 000, 1–?? (2017) Gravitational instability of gas-dust mixture 17

The spatially harmonic solution of equations (A9-A12) corre- sponding to exponential growth of perturbations with the growth 2 ′ 1/3 ωδ˜ = Vˆ k δ˜ + √2τ h˜ . (B6) rate equal to (fτ) ωff reads · ′ Using the variational principle valid for modes of perturba- h , δ cos k r∗ and ug , gp sin k r∗. (A13) ∝ · ∝− · tions with the amplitude constant in time, which is also known as It is worth noting that δ > 0 means not just the increase of the the method of Whitham (2011), it is possible to derive the energy dust density as compared to its background value, but rather the in- of gas-dust wave from its fundamental symmetry to translations in crease of an excess of the dust density compared to the gas density, time. ′ 2 ˜ ˜ δp > h /cs , in the course of collapse. This excess is continuously The averaged Lagrangian, L(h, δ), reads generated by perturbation of the relative velocity of dust with re- ′ 2 2 2 2 2 (h˜ ) 2 δ˜ spect to gas, which, in turn, arises due to gravitational attraction L =(ω k + τ ) + √2fτ h˜′δ˜ f(ω Vˆ k) . (B7) 2 || 2 of grains into the potential well of gas overdensities, see equation − − − (A8). This is the way how an incremental gravitational potential, This Lagrangian provides equations (B1-B2) equivalent to the Euler-Lagrange equations Φp, is produced. If it were not for the drift of the grains through the gas, the non-zero Φp would not have appeared. The corresponding ∂L = 0, additional gravitational acceleration stimulates contraction of the ˜′ ∂h (B8) gas which further increases gas density and the accumulation rate ∂L of the excess dust. As compared to the long-wavelength as well as = 0. ∂δ˜′ short-wavelength limits, dust destabilises the medium much more effectively because of the most favourable conditions for drift, cf. It can be seen that L = 0 provided that h˜1 and δ˜ satisfy equa- equations (A2), (A4) and (A11). tions (B5-B6). Accordingly, the wave energy is as follows 2 ∂L ∂θ ∂L 2 2 δ˜ E L = ω = ω h˜1 fω . (B9) ≡ ∂(∂θ/∂t) ∂t − ∂ω − 2 APPENDIX B: ENERGY OF MODES OF GAS-DUST As f 0, eq. (B9) shows that the energy of HSW and SDW PERTURBATIONS IN THE PARTICULAR CASE is, respectively,→ positive and negative definite. Indeed, the case of ASSOCIATED WITH THE MODE COUPLING HSW corresponds to δ˜ 0, while ω (k2 τ 2)1/2 provided → → − The reduced equations describing resonance of two modes in the that k>τ. On the other hand, the case of SDW corresponds to ˜′ ˆ vicinity of the mode crossing can be obtained from equations (20- h 0, while ω V||k, which confirms that the energy SDW is → → 22) by setting ∂t (V ) in the leading order in small f 1 negative each time the projection of the drift velocity, Vˆ , onto the ≈ ·∇ ′ ≪ and omitting the terms fh in equation (20) and all terms f wavevector of SDW is positive. in equation (21) after equation∼ (22) has been used there to express∼ The existence of HSW and SDW having energies of the op- v. One finds posite signs at the mode coupling allows for the standard physical ∇ · 2 ′ ′ explanation of instability of the gas-dust mixture in this particu- 1 ∂ h 2 ′ 2 h 2 f 2 2 = h + ωff 2 + fωff δ (V )δ, (B1) lar case. Following Cairns (1979) and recently Zhuravlev (2019) c ∂t ∇ c − ts ·∇ s s the growing (or damping) coupled mode of gas-dust perturbations may be considered as the resonant coalescence of HSW and SDW, ′ ′ ∂δ h 2 h which provides an exchange with energy between the waves. The +(V )δ +(V ) 2 ts ωff 2 = 0. (B2) ∂t ·∇ ·∇ cs − cs energy of coupled mode is conserved, while the energy flow be- 8 tween the coalescing HSW and SDW provides the growth (damp- An additional condition , ing) of their amplitudes as long as the energy flows from the nega- 2 (V )= ts ωff , (B3) tive (positive) energy wave to the positive (negative) energy wave. ·∇ As the strict condition of the mode coupling (B3) is not true, is imposed on terms that make up the coupling term in the disper- there is no simple way to construct the Lagrangian for modes of sion equation, i.e. on the terms δ in equation (B1) and the terms perturbations. As soon as the coupling term is complex, neutral ′ ∝ h in equation (B2). The condition (B3) reproduces to the mode ∝ modes do not exist for f > 0 at any wavenumber. Moreover, ac- coupling of HSW and SDW as it is a more restrictive analogue of cording to the Lagrangian theory of perturbations developed for the condition (55), see Section 5.3. single-fluid dynamics, see Friedman & Schutz (1978), the energy The particular solution of equations (B1-B2) with the condi- of growing (damping) modes must vanish. That is why, there is tion (B3) is taken in the following way no straightforward generalisation of the standard interpretation of ′ 2˜′ the resonant instability given here onto arbitrary ratio between τ h = csh cos(θ π/4), − (B4) and Vˆ . Note that the coupling term is real within the ’standard’ ˜ || δ = δ cos θ, concept of resonance between modes, see Fabrikant et al. (1998). where As f > 0, the mode crossing is replaced either by the mode cou- pling leading to instability or by the avoided crossing, which keeps θ = ωt∗ + k r∗. the modes neutral (see Zhuravlev (2019) for applications to the − · dynamics of gas-dust perturbations in protoplanetary disc). In the The tilded quantities satisfy the following equations: model of Zhuravlev (2019) the coupling term becomes complex 2 2 2 2 ω h˜′ = k h˜′ τ h˜′ fτ √2 δ,˜ (B5) in the next order over the small stopping time turning one of the − − neutral modes taking part in the avoided crossing into a growing one. The corresponding mechanism of instability was referred to 8 ˆ 2 in the dimensionless units, this is V||k = τ as ’quasi-resonant’ to distinct it from the ’standard’ resonant case.

MNRAS 000, 1–?? (2017) 18 V.V. Zhuravlev

At least technically, the mechanism of the instability considered in this work also goes beyond the ’standard’ case of the mode cou- 0,02 pling, however, such a terminological distinction is not used in this study.

0,01

Re APPENDIX C: REMARKS ON THE 0 LONG-WAVELENGTH HS18 INSTABILITY

The solution (67) along with the limit (68) correspond to the fol- -0,01 lowing restrictions

-0,02 0,0001 0,001 0,01 0,1 1 10 100 1 ω k & kVˆ (C1) k ≫ ≫ || along with 2 kVˆ fkVˆ ω . (C2) || ≫ || ≫ 0,002 Assuming ωff = 0, while ts/tev ω and csts/lev k, where ∼ ∼ tev and lev are, respectively, the characteristic time and scale of dy- 0

namics of the perturbed flow, equation (22) shows that perturbation Im of the relative velocity is determined by the acceleration of gas, i.e. -0,002 v ω2h′. In turn, this means that the terms 2h′ and f v ∇ · ∼ ∇ ∼ ∇ on the RHS of equation (20) can be omitted as they are small com- -0,004 pared to the main term on its LHS. Thus, the acceleration of gas is determined by the aerodynamic drag emerged from the excess (de- -0,006 ficiency) of dust drifting through the gas at the velocity of the bulk -0,008 drift, V. Therefore, in the long-wavelength limit, both gas and dust 0,0001 0,001 0,01 0,1 1 10 100 behave like pressureless fluids interacting via aerodynamic drag, k which was recognised by HS18. Inequality (C2) applied to the re- duced equation (20) indicates that, additionally, h′ δ, i.e. pertur- ≫ Figure C1. The curves in top and bottom panels show, respectively, ℜ[ω] bation of the dust density, δp, is mostly generated by the compres- ℑ sion (expansion) of gas. Moreover, inequality (C2) guaranteers that and [ω], where ω is the solution of equation (28) obtained for τ = 0 and Vˆ . . Solid and dotted lines show two SW and SDW for f , v can be omitted in equation (21) as well. Thus, according to || = 0 5 = 0 whereas short- and long-dashed lines and dot-dashed line show the corre- the∇ · reduced equation (21), perturbation of the dust fraction is deter- sponding solutions for f = 0.01. mined solely by perturbation of the gas density. One arrives at the following set of equations for the long-wavelength HS18 instability ∂u V oscillations of gas velocity perturbation are not synchronised with g = f δ, (C3) ∂t ts the force driving these oscillations. Indeed, for the ordinary SW, equations (C3) and (C5) are replaced by the single equation ′ 1 ∂h ∂ug ′ 2 = ug , (C4) = h , (C6) cs ∂t −∇ · ∂t −∇ ′ which leads to the phase difference between ug and h equal −∇ ∂δ 1 ′ to π/2. But this is not the case as the driving force is described by = 2 (V )h . (C5) ∂t − cs ·∇ equations (C3) and (C5). Equation (C5) shows that variations of the dust fraction Let the gas be moving along the x-axis only, which at the same emerge due to the bulk drift of the dust pre-compressed (pre- time be the direction of the bulk drift of the dust. As the growing expanded) solidly with the gas, which is contrasted to the situa- branch corresponding to tion when the dust clumps due to the relative motion of gas and i+ √3 1 3 2 3 dust, which is excited by perturbation of the gas pressure gradi- ω = f / (Vˆ k) / 2 || ent. The latter is the case for GI of the gas-dust mixture discussed in this paper, see Appendix A, as well as for the subsonic RDI is considered, it can be shown that δ cos(ϕ π/6) and ug ′ ∝ − ∝ in the rotating gas-dust mixture of protoplanetary discs, see e.g. cos(ϕ + π/6), provided that h cos ϕ, where ϕ is the phase ∝ Squire & Hopkins (2018a) and Zhuravlev (2019). of oscillations, see also HS18. The immediate cause for growth of As the wave propagates through the medium with gas-dust the wave amplitude is the non-zero net work of the driving force perturbations described by equations (C3-C5), perturbation of the acting onto the fluid elements. On the simple background given by gas density is generated by the divergence of its velocity perturba- equations (7-10) the net work of the driving force reads tion, which is similar to the ordinary SW. However, the feedback V is different as compared to SW. Namely, the gas velocity perturba- A = f δ ugdt cos(ϕ π/6) cos(ϕ + π/6)dϕ = π/2 ts ∝ − tion is generated by the perturbation of the gas pressure gradient I I indirectly through perturbation of the dust fraction. Accordingly, over the oscillation period of the gas element.

MNRAS 000, 1–?? (2017) Gravitational instability of gas-dust mixture 19

On the contrary, in the ordinary SW It is also possible to take into account the next order correction

in Vˆ|| to estimate (C13). It emerges due to the non-zero divergence ′ A = h ug dt sin ϕ cos ϕ = 0. of gas velocity perturbation induced by the oscillations of the dust − ∇ ∝ I I density. As far as the main order correction to the frequency of Thus, the driving force δ provides the transition of the back- SDW is small, v remains to be negligible for the divergent gas ∝ ground energy of the dust drift into the energy of wave. flow as well. This∇ · can be seen from equation (22), where it is taken For the subsonic dust drift, Vˆ < 1, the HS18 instability ceases ˆ 2 into account that ∂t (V ) fV|| in the dimensionless form. as k approaches unity. An exact solution of the general equation Hence, the only new− term·∇ which∼ needs to be included stands on (28) taken for the non-self-gravitating medium is shown in Figure the LHS of equation (20). Accordingly, the improved estimate of C1. It approaches (67) in the long-wavelength limit, while each of damping SDW is the following the three branches seen in Figure C1 become damping at its own ˆ 2 constant rate in the limit of high k. The top panel in Figure C1 V|| ω kVˆ if . (C14) shows that the two curves introducing the long-wavelength HS18 ≈ || − ˆ 2 1 V|| instability approach the SW dispersion relation, while the third one, − which is damping for all k, approaches the SDW dispersion rela- It can be checked that equation (C14) is in good agreement with tion. In order to analytically reproduce the constant damping rates an accurate asymptotics at high k, see the dot-dashed curve at the at k , one should treat the first two solutions (the dashed lines bottom panel of Figure C1. in Figure→∞C1) as SW propagating on the dust-laden background with additional bulk drift of the dust, while the third solution (the dot-dashed curve in Figure C1) as SDW exciting the subsonic os- cillations of gas. Let the frequency of gas-dust wave be ω k 1. Provided ≈ ≫ ω ts/tev, RHS of equation (22) vanishes and it yields ∼ v ug, (C7) ≈− i.e. in the high-frequency dust-laden SW the dust velocity pertur- bation is negligible, up 0, because of the enhanced inertia of the grains. According to equation≈ (21), this means that h′ δ 2 . (C8) ≈− cs With help of relations (C7) and (C8) equation (20) is ex- pressed as

2 ′ ′ ∂ h 2 2 ′ f ∂h ′ 2 = cs h +(V ) h , (C9) ∂t ∇ − ts ∂t ·∇   which yields the following approximate solution for f 1: ≪ if ω k 1 Vˆ . (C10) ≈± − 2 ± || It can be checked that equation (C10) is in good agreement with an accurate asymptotics at high k, see the dashed curves in the bottom panel of Figure C1. Now, let the spatially periodic perturbations of the dust density be advected by the background drift of the dust. Assuming that Vˆ|| 1 implies that the corresponding frequency of oscillations of aerodynamic≪ dust feedback introduced by the last term on the RHS of equation (20) is small compared to the frequency of SW with the same wavelength. Thus, the problem may be considered in the ′ 2 limit of ug 0, ∂th /c 0, which implies that, according ∇ · → s → to equation (22), v 0 and equation (20) is reduced to ∇ · → 2 f h′ (V )δ. (C11) ∇ ≈ ts ·∇ In the same limit, equation (21) reads

∂δ 1 ′ +(V )δ 2 (V )h . (C12) ∂t ·∇ ≈− cs ·∇ Equations (C11-C12) give the following solution for modes of per- turbations 2 ω kVˆ ifVˆ , (C13) ≈ || − || which is valid up to the leading order in the Vˆ 1. ≪ MNRAS 000, 1–?? (2017)