VALOR DEL TERRENO

As previously explained in Chapters4and5, dynamical instabilities arise naturally in wind-cloud interactions and they not only deform the cloud but also alter the morphology of the associated filaments. Previous studies by e.g.,Klein et al.(1994);

Gregori et al.(2000);Pittard & Parkin(2016) showed that four instabilities can have significant effects on the formation and evolution of wind-swept clouds. These are the KH, RT, Richtmyer-Meshkov (hereafter RM), and tearing-mode (hereafter TM) instabilities.

As described in Section 5.2, the first type of these perturbations, i.e., the KH instability results from shearing motions occurring at the boundary layer separating filament and ambient gas (see e.g., the 3D study of the KH instability byRyu, Jones & Frank 2000). In the models presented here, the shear layer emerges from the velocity difference across the wind-cloud interface (see Section5.2.2and references therein for a complete picture of the KH instability). The sinuosity (i.e., the ripples) observed at the lateral boundaries of the filamentary structures in the panels of Figure 5.3 is caused by the KH instability. In the classical approximation, the growth time-scale of the KH perturbations is given by

(6.1) tKH tcc ' " ⇢0c⇢0wk2KH (⇢0c+⇢0w)2(v 0 w v0c)2 2B0₂ k2 KH (⇢0c+⇢0w) # 1 2 Mwcw 2rc 12 ,

where the primed quantities represent the values of the physical variables at the location of shear layers, andkKH = 2_KH⇡ is the wavenumber of the KH perturbations

(see Chapter XI ofChandrasekhar 1961). While the KH instability is shear-driven as mentioned above, the second type of perturbations, i.e., the RT instability is buoyancy-driven in fluids with different densities (i.e., no shearing motions are needed to trigger it). In fact, as described in Section5.2, the RT instability arises

6.3. FILAMENTS IN DIFFERENT ENVIRONMENTS

when an initial perturbation at the interface between the leading edge of the cloud and the impinging wind is allowed to grow under the influence of the wind-driven acceleration (see also Stone & Gardiner 2007). The RT instability originates a collection of low-density bubbles (of wind gas) and high-density spikes (of cloud material) at the leading edge of the clouds, which grow to disrupt these clouds at late times in the evolution (see Section5.2.4and references therein for a complete picture of the RT instability). In the classical theory, the growth time-scale of the RT perturbations is given by (6.2) tRT tcc ' " ⇢0_c ⇢0_w ⇢0c+⇢0w ! akRT 2B02_k2 RT (⇢0c +⇢0w) # 1 2 Mwcw 2rc 12 ,

where the primed quantities represent the values of the physical variables at the leading edge of the cloud,ais the local, effective acceleration of dense gas, andkRT =

2⇡

RT is the wavenumber of the RT perturbations (see Chapter X ofChandrasekhar

1961). Note that Equations (6.1) and (6.2) correspond to classical analyses of the instabilities in the incompressible regime. Therefore, the values provided by them should be considered as indicative numbers for the growth time-scales of the KH and RT instabilities in the compressible cases under analysis here. Table6.2

provides reference time-scales for the growth of KH and RT instabilities, estimated from Equations (6.1) and (6.2) using simulation results as input quantities.

Note that in models with transverse magnetic field components, a thin magnetic layer envelops the cloud and provides additional stability to the wind-cloud inter- face (see e.g.,Asai et al. 2004,2005;Dursi & Pfrommer 2008). Therefore, the above Equations (6.1) and (6.2) only provide lower limits for the growth times of these instabilities in those models. In order to properly account for the stabilising effects of this draping magnetised layer on the growth of dynamical instabilities,Dursi

(2007) modified the classical equations ofChandrasekhar(1961) by introducing a three-layer configuration (see Figure 1 in their paper). Thus, the reader is referred to this work in case further comparisons are desired. Dursi(2007) provides ex- pressions for the growth rate of KH and RT instabilities in the presence of a thin magnetic layer in Equations (30) and (35) in their paper.

The third type of perturbations, i.e., the RM instability (Richtmyer 1960;Meshkov 1969) grows at the beginning of the wind-cloud interaction as a result of the impulsive acceleration produced by the refraction of the initial shock wave into the cloud (seeSano et al. 2012;Sano, Inoue & Nishihara 2013for recent studies). The RM instability is shock-driven and can be considered as an impulsive RT instability

CHAPTER 6. FILAMENT FORMATION IN UNIFORM MEDIA

(see e.g.,Kull 1991). In fact, the growth mechanism of both instabilities is similar as RM perturbations also emerge from the vorticity deposited onto the corrugated shock-cloud interface by the misalignment of pressure and density gradients in both media (seeGrove et al. 1993;Brouillette 2002).

Several morphological features (such as bubbles and spikes) and late-stage turbu- lent mixing are all characteristic of both the RT and RM instabilities (seeDimonte 1999;Khan et al. 2011), however, the exponential growth rate of the RT modes makes the linearly-growing RM instability only important at the very early stages of the evolution of wind-cloud systems (see e.g.,Gregori et al. 2000;Nakamura et al. 2006;Pittard & Parkin 2016). This effect combined with the smoothed edges imposed on the spherical clouds employed in the models presented here minimise the role of the RM instabilities (with respect to KH and RT perturbations) in the disruption of clouds and subsequent formation of filaments.

In MHD models, a fourth perturbation emerges, namely the TM instability (see

Furth, Killeen & Rosenbluth 1963;Parker 1979; Chapter 20 ofGoldston & Rutherford 1995). The TM instability grows when oppositely directed magnetic field lines are pushed together, so that they reconnect to form closed magnetic islands (see a description inLazarian & Vishniac 1998). This type of perturbations is particularly relevant for wind-cloud models where the magnetic field lines are transverse to the direction of the flow (e.g., in models MHD-Tr, MHD-Ob, MHD-Ob-S, and MHD-Ob-I). In these scenarios, the transverse magnetic field lines wrap around the cloud converging behind it with oppositely-directed vector fields (see Section

6.3.4 below). This convergence forms a long and thin current sheet along the transition layer at the rear side of the cloud.

As the lines are brought into contact by the wind flow, the newly-formed transition layer becomes unstable to resistive TM modes and is broken up into several mag- netic islands (seeMelrose 1986;Biskamp 1993). As a result, the local topology of the magnetic field changes and Alfvén waves are triggered from the reconnection site (see e.g.,Lazarian & Vishniac 1998). Magnetic reconnection via TM instabilities has been reported in previous studies of wind/shock-cloud systems (e.g., see the work byJones et al. 1996;Miniati et al. 1999a,b, so this phenomenon is also expected to play an important role in shaping the magnetic field topology of the wind-swept clouds (and filaments) presented here. Note also that, in the models presented here, the dissipation caused by the numerical approximations and truncation errors of the scheme mimics resistivity and allows magnetic reconnection to happen (see also a discussion in Section 2.2 ofJones et al. 1996and Section 3 ofMiniati et al. 1999b).

6.3. FILAMENTS IN DIFFERENT ENVIRONMENTS

The morphology, endurance, and magnetic properties of filaments are determined by the growth rates of the aforementioned dynamical instabilities. As I show in the following sections, the growth of these perturbations at fluid boundaries heavily depend upon: 1) whether or not the medium in which they emerge is magnetised, and 2) how the magnetic field is initially oriented when present. Consequently, it is convenient to describe some details of the evolution in each M/HD model independently.