In this section I study the effects of numerical resolution (defined by the number of grid cells per cloud radius) on the 2D HD simulations presented in this chapter. In order to study how my choice of standard resolution (i.e.,R256or256cells per
cloud radius) affects the diagnostics calculated in 2D HD wind-cloud simulations, I report a set of five runs (see Table4.3) with the same initial conditions as model 2dHDu, but different resolutions ranging from32to512cells per clouds radius. Figure4.7shows the time evolution of two diagnostics, namely the lateral elong- ation of the cloud (◆1,cloud, see Equation3.17) in Panel A1, and the total mass flux
(i.e., cloud plus wind material),F, through a line located atX2,cut = +6rc in Panel
B1 (see Equation3.21). The curves shown in this figure correspond to a single HD simulation: 2dHDu (see the initial conditions in Table4.1), run at different resolutions. Panel A1 shows convergence within5 %in all cases, except for theR32
model, untilt/tcc =0.75(note that theYaxis in the plot is logarithmic). After this
time, the clouds in the low-resolution models (i.e.,R32,64) are less confined than
their high-resolution counterparts. This is caused by the wind-cloud boundary layers being under-resolved at low resolutions. Thus, when the clouds expand in these models, they appear⇠ 10 % more elongated at late times than their high- resolution counterparts. Based on this, I conclude that resolutionsR>64are needed
to adequately capture the late expansion phase and break-up of the cloud.
In addition, Panel B1 of Figure4.7shows the flux of (cloud plus wind) mass flowing through a line located at X2,cut = +6rc, perpendicular to theX2 axis. I find that
all the simulations are well converged for this parameter, except the ones at low
4.5. RESOLUTION STUDY IN TWO DIMENSIONS 0.1 1 10 0 0.25 0.5 0.75 1 1.25 ι1,cloud t / tcc A1) Lateral Elongation
R32 R64 R128 R256 R512 0 0.5 1 1.5 2 0 0.25 0.5 0.75 1 1.25 F / F w,0 t / tcc B1) Mass Flux R32 R64 R128 R256 R512
Figure 4.7 Comparison of the lateral elongations (Panel A1) and mass fluxes (Panel B1) measured in 2D HD models with different resolutions, ranging fromR32throughR512. All these models have the same initial conditions as model 2dHDu. The curves show convergence within5 % for all the resolutions, except for theR32,64runs, which poorly resolve the wind-cloud/filament interfaces. ResolutionsR>64capture the dynamical instabilities better as revealed by the appear- ance of stronger spikes in the higher-resolution curves of Panel B1. The stochasticity of these spikes shows the chaotic nature of the evolution after the onset of turbulence.
resolutions, i.e.,R32,64, which fail to capture turbulence arising as cloud gas mixes
with the ambient medium downstream. In the figure this is revealed by the lack of spikes (associated with turbulent motions) in the curves of these runs. On the other hand, the curves representing high-resolution simulations (R>64) show
clear evidence of random vortical motions, which indicate that wind-filament interfaces have been resolved adequately. Based on the behaviour of the curves presented in both Panels A1 and B1 of Figure 4.7, I conclude that resolutions greater than64cells per cloud radius are needed to study wind-cloud interactions in the configuration given in Table4.1. Note also that after the cloud breaks up (att/tcc⇠1), turbulence dominates and the evolution becomes more chaotic than
before (see e.g., the divergence of curves in Panels A1 and B1 att/tcc=1.1).
The aforementioned effects of the numerical resolution on the evolution of clouds and filaments can also be seen in Figure4.8, which shows 2D snapshots of the normalised mass density in model 2dHDu at increasing resolutions. The snapshots correspond to timest/tcc =0.4(Panel A) andt/tcc=1.0(Panel B). I find thatR32is
dominated by diffusion and fails to capture the growth of dynamical instabilities and turbulence. A resolution ofR64 captures well the instabilities at the leading
edge of the cloud, but it fails to produce the turbulent tail seen in higher-resolution models. In fact, the late-stage snapshots indicate that the downstream turbulence (in the filamentary tail) is only captured at resolutions greater than or equal toR128.
CHAPTER 4. CYLINDRICAL CLOUDS IN UNIFORM MAGNETIC FIELDS
R32 R64 R128 R256 R512
Figure 4.8 2D snapshots of model 2dHDu at different resolutions (increasing from left to right), showing the logarithmic mass density of the cloud (normalised with respect to its initial value). The snapshots correspond to timest/tcc=0.4(Panel A) andt/tcc=1.0(Panel B). Even though the break-up of the cloud seems well resolved at resolutions ofR64, turbulence is only captured at resolutions greater or equal thanR128as evidenced by the structure of the filament at both times. As a result, this resolution ofR128is optimal for capturing both the break-up process and the turbulence in the tail.
atively similar to that of modelR128, so I conclude that the optimal resolution for
studying 2D wind-cloud systems in the parameter space mentioned above is128
cells per cloud radius. This is in agreement to what was found (i.e., convergence at resolutions⇠R100) byKlein et al.(1994);Nakamura et al.(2006) in their 2D and
2.5D simulations for lower density contrasts and higher Mach numbers than the models presented here.
Note that the inclusion of magnetic fields and source terms, such as radiative cool- ing or thermal conduction, in wind-cloud simulations could alter this expectation for the optimal resolution (see e.g.,Yirak et al. 2010, who concluded that conver- gence in models with radiative cooling could only be reached when the cooling layers behind shocks are adequately resolved). Therefore, this result cannot be, in