[ Article ]
Journal of the Korean Astronomical Society - Vol. 53, No. 2, pp.49-57
ISSN: 1225-4614 (Print) 2288-890X (Online)
Print publication date 30 Apr 2020
Received 10 Oct 2019 Accepted 18 Mar 2020

# DECAY OF TURBULENCE IN FLUIDS WITH POLYTROPIC EQUATIONS OF STATE

JEONGHOON LIM1 ; JUNGYEON CHO1, 2
1Department of Astronomy and Space Science, Chungnam National University, 99, Daehak-ro, Yuseong-gu, Daejeon 34134 jhlim0918@o.cnu.ac.krjcho@cnu.ac.kr
2Korea Astronomy and Space Science Institute, 776, Daedeokdae-ro, Yuseong-gu, Daejeon 34055

Correspondence to: J. Cho

Published under Creative Commons license CC BY-SA 4.0

## Abstract

We present numerical simulations of decaying hydrodynamic turbulence initially driven by solenoidal (divergence-free) and compressive (curl-free) drivings. Most previous numerical studies for decaying turbulence assume an isothermal equation of state (EOS). Here we use a polytropic EOS, Pργ , with polytropic exponent γ ranging from 0.7 to 5/3. We mainly aim at determining the effects of γ and driving schemes on the decay law of turbulence energy, Etα. We additionally study probability density function (PDF) of gas density and skewness of the distribution in polytropic turbulence driven by compressive driving. Our findings are as follows. First of all, we find that even if γ does not strongly change the decay law, the driving schemes weakly change the relation; in our all simulations, turbulence decays with α ≈ 1, but compressive driving yields smaller α than solenoidal driving at the same sonic Mach number. Second, we calculate compressive and solenoidal velocity components separately and compare their decay rates in turbulence initially driven by compressive driving. We find that the former decays much faster so that it ends up having a smaller fraction than the latter. Third, the density PDF of compressively driven turbulence with γ > 1 deviates from log-normal distribution: it has a power-law tail at low density as in the case of solenoidally driven turbulence. However, as it decays, the density PDF becomes approximately log-normal. We discuss why decay rates of compressive and solenoidal velocity components are different in compressively driven turbulence and astrophysical implication of our findings.

## Keywords:

ISM: general, hydrodynamics, turbulence

## 1. INTRODUCTION

Supersonic turbulence in the interstellar medium (ISM) is a well-known phenomenon and plays an essential role in star formation processes (Larson 1981; Padoan & Nordlund 2002; Mac Low & Klessen 2004). Given that driving mechanisms of astrophysical turbulence are usually intermittent in both space and time, it is natural for turbulence to decay. Earlier studies showed that non-driven turbulence decays quickly in approximately one large-eddy turnover time (for hydrodynamic turbulence, see e.g., Lesieur 2008; for magnetohydrodynamic turbulence, see Mac Low et al. 1998; Stone et al. 1998), which is consistent with the fact that an energy cascade occurs within one large-scale eddy turnover time even in the case of strongly magnetized turbulence (Goldreich & Sridhar 1995).

It has been analytically suggested that turbulence energy decays with a power-law of the form Etα (see e.g., Chapter 7 of Lesieur 2008). Results from previous numerical studies of turbulence have converged to a value of α of approximately unity, and it does not strongly depend on the degree of magnetization and compressibility (Mac Low et al. 1998; Stone et al. 1998; Biskamp & Müller 1999; Ostriker et al. 2001; Cho et al. 2002).

Even if the consensus that turbulence quickly decays has been numerically established for the last two decades, the previous numerical results rely heavily on the isothermal condition. However, as long as various density and temperature phases in the ISM (Ferrière 2001) are concerned, the use of a polytropic equation of state (EOS)

 $P=K{\rho }^{\gamma },$ (1)

where P is the pressure, ρ is the density, and both K and γ are constants, is a valid approach (see Vazquez-Semadeni et al. 1996 and references therein). The polytropic EOS has been used for many astrophysical problems, such as complex chemical processes (Spaans & Silk 2000; Glover & Mac Low 2007a) or turbulence (Scalo et al. 1998; Li et al. 2003; Glover & Mac Low 2007b; Federrath & Banerjee 2015).

Besides a variety of density and temperature phases, a wide range of driving agents of turbulence also characterizes interstellar turbulence (see Federrath et al. 2017 for a review). Based on its compressibility, we may consider two extreme types of driving: solenoidal (divergence-free) and compressive (curl-free). Until recently, solenoidal driving had been mainly used for turbulence studies. However, Federrath et al. (2010) showed that compressive driving and solenoidal driving can have different statistics. For example, they showed that “the former yields stronger compression at the same RMS Mach number than the latter, resulting in a three times larger standard deviation of volumetric and column density probability distributions.” To the best of our knowledge, scaling relations of decaying polytropic turbulence initially driven by compressive driving have not been studied yet.

The main goal of this paper is to examine whether the decay exponent α depends on the value of the polytropic exponent γ. Here we concentrate on decay of polytropic turbulence driven by either solenoidal or compressive driving in both transonic and supersonic regimes. Hence, we expect to demonstrate how polytropic EOS and types of driving affect decaying turbulence. In addition, we also investigate the probability density function (PDF) of gas density and skewness of the PDF in decaying polytropic turbulence initially driven by compressive driving.

The paper is organized as follows. We explain our motivation and numerical method in Section 2, and present the results from our numerical simulations in Section 3. We discuss our finding and its astrophysical implication and give a summary in Section 4.

## 2. MOTIVATION AND NUMERICAL METHOD

### 2.1. Motivation

As we described earlier, decay of solenoidally driven isothermal turbulence follows Etα with α ≈ 1. The type of driving or γ may affect this scaling relation.

First, compressive driving yields more compressions at the same Mach number. Therefore, while decaying, compressed regions could generate additional kinetic energy via expansion, which could affect the rate of turbulence decay.

Second, regarding the effects of γ on decaying turbulence, only limited regions of the parameter space have been studied. Mac Low et al. (1998) found that supersonic turbulence with γ = 1.4 decays with α ~ 1.2. For isothermal cases (i.e., γ = 1), they found that α is nearly unity. This suggests that the scaling exponent α in Etα only weakly depends on γ as assumed by Davidovits & Fisch (2017). However, Mac Low et al. (1998) used a random initial velocity perturbation which follows a power-law, and a constant initial density. Therefore, it is necessary to test the decay law using initial velocity and density data cubes from actual turbulence simulations with both soft EOS (i.e., polytropic γ < 1) and stiff EOS (i.e., polytropic γ > 1).

Third, the effects of γ and the type of driving on density PDF of turbulence have been addressed in several previous studies. For example, Federrath et al. (2010) showed that solenoidal driving and compressive driving can produce different statistics of isothermal turbulence. In addition, Federrath & Banerjee (2015) found that the density PDF of solenoidally driven turbulence with polytropic γ = 5/3 has a power-law tail at low density, which is not observed in isothermal turbulence. However, earlier studies have not addressed turbulence with polytropic EOS and compressive driving. In this paper, we use both compressive driving and polytropic EOS to investigate density PDF and skewness of driven and decaying turbulence.

### 2.2. Numerical Method

2.2.1. Numerical Code

We use an Essentially Non-Oscillatory (ENO) scheme (see Cho & Lazarian 2002) to solve the ideal hydrodynamic equations in a periodic box of size 2π (≡ L0):

 $\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho \mathrm{v}\right)=0,$ (2)
 $\frac{\partial \mathrm{v}}{\partial t}+\mathrm{v}\cdot \nabla \mathrm{v}+{\rho }^{-1}\nabla P=\mathrm{f},$ (3)

where f is a driving force, P is the pressure (see Section 2.2.2), ρ is the density, and v is the velocity. The density and velocity are set to 1 and zero at t = 0 to assume a static medium with a constant density at the beginning.

2.2.2. Simulations

We use 5123 grid points in our periodic computational box. The peak of energy injection occurs at k ~ kf , where k is the wavenumber and kf (≈ 6 or 8) is the driving wavenumber at which the energy injection rate peaks. Therefore, in our simulations, we generate turbulence approximately at a scale kf times smaller than the computational box. We drive turbulence in Fourier space using 100 forcing components, which are nearly isotropically distributed in the range kf /1.3 ≾ k ≾ 1.3kf . We consider solenoidal (∇ · f = 0) and compressive (∇ × f = 0) drivings. In the former driving, we extract forcing components normal to k in Fourier space and carry out an inverse Fourier transform to produce them in real space. In the latter driving, the procedure is the same, but we extract forcing components parallel to k in Fourier space.

In both drivings, the driving vectors continuously change with a correlation time comparable to the large-eddy turnover time. We also adopt a polytropic EOS

 $P={P}_{0}{\left(\frac{\rho }{{\rho }_{0}}\right)}^{\gamma }=\left(\frac{{c}_{s0}^{2}{\rho }_{0}}{\gamma }\right){\left(\frac{\rho }{{\rho }_{0}}\right)}^{\gamma }$ (4)

where P is the normalized pressure, and P0, cs0, and ρ0 are the initial pressure, sound speed, and density, respectively. The sonic Mach number Ms is defined by

 ${M}_{s}\equiv \frac{{\upsilon }_{\mathrm{r}\mathrm{m}\mathrm{s}}}{{c}_{s0}},$ (5)

where vrms is the rms velocity. We vary γ and the sonic Mach number Ms to consider both soft and stiff EOS in transonic and supersonic regimes.

Table 1 lists our simulation models. We use the notation XMSY-γZ, where X = S or C refers to solenoidal (S) or compressive (C) driving; Y = 1, 3, or 5 refers to the sonic Mach number Ms; Z = 0.7, 1.0, 1.5, or 5/3 refers to the value of γ. We keep driving turbulence until the system reaches saturation, after which the driving is turned off to let turbulence freely decay. In decaying simulations, time is normalized like t = tcode/ted. Here, tcode is the time in code units, ted = (L0/kf )/v0 is the large-eddy turnover time, and v0 is the velocity at the moment turbulence starts decaying.

Simulation runs and parameter values

## 3. RESULTS

### 3.1. Decay of Hydrodynamic Turbulence with Polytropic EOS

3.1.1. Decay of Turbulence Driven by Solenoidal Driving

Here, we consider decaying polytropic turbulence initially driven by solenoidal driving. Figures 1 and 2 show the decay of <v2>1 and the standard deviation of density fluctuations σρ/ρ0 , respectively, where <· · ·> denoted spatial averaging. From left to right, the sonic Mach number Ms is ~ 1, ~ 3, and ~ 5, respectively. Blue, red, cyan, and green curves in each panel correspond to γ of 0.7, 1.0, 1.5, and 5/3, respectively.

Time evolution of <v2> in decaying turbulence initially driven by solenoidal driving with γ = 0.7 (blue), 1.0 (red), 1.5 (cyan), or 5/3 (green), where <· · ·> indicates spatial averaging. Left: Ms ~ 1. Center: Ms ~ 3. Right: Ms ~ 5. Turbulence starts decaying at tcode/ted = 0. We normalize <v2> by υ02 which is the value at tcode/ted = 0. The black dotted lines in all three panels are reference lines for different power-law exponents.

Same as Figure 1, but for the decay of the standard deviation of the density fluctuation σρ/ρ0 . We normalize σρ/ρ0 by (σρ/ρ0)0 which is the value at tcode/ted = 0.

First of all, we clearly see that the decay of <v2> follows a power-law of the form <v2> ∝ tα, and α is almost the same at the same Ms regardless of the value of γ (see Table 1 for measurements of α). The decay is steeper in the case of Ms ~ 1 (α ~ 1.2) than in supersonic cases (α ~ 1.0). Second, similar to the case of <v2>, the decay of σρ/ρ0 is hardly affected by γ. In addition, except for the case Ms ~ 1 (see Table 1), the power-law exponent for σρ/ρ0 is about half of that for <v2> at the same Ms. For the cases with γ = 1 (i.e., isothermal cases), this result is consistent with the fact that the standard deviation of density fluctuations scales approximately linearly with the sonic Mach number (e.g., Padoan et al. 1997; Passot & Vázquez-Semadeni 1998), which implies σρ/ρ0Ms ∝ <v2>1/2tα/2 . Our results imply that a similar argument holds true for γ ≠ 1.

3.1.2. Decay of Turbulence Driven by Compressive Driving

We now deal with the decay of polytropic turbulence initially driven by compressive driving. Figures 3 and 4 show the decay of <v2> and σρ/ρ0 , respectively. As in the case of solenoidal turbulence,2 we use different values of Ms (from left to right panels) and γ (curves with different colors).

Same as Figure 1, but for compressively driven turbulence. Note that unlike Figure 1, <v2> slightly increases along the decay as indicated by a black arrow in each panel; this is most apparent in the case of Ms ~ 1 (left panel).

Same as Figure 2, but for compressively driven turbulence.

Similar to the result from solenoidal turbulence, both <v2> and σρ/ρ0 in compressively driven turbulence exhibit power-law decay, and γ hardly affects the decay rate (see Table 1 for measurements of α). According to Figure 3, the power-law exponent α is ~1.0 for Ms ~ 1, and ~0.8 for Ms > 1, which means that the decay of compressively driven turbulence is slower than that of solenoidal turbulence at the same Ms. As can be seen from Figure 4, the power-law exponent α for the decay of σρ/ρ0 is about half of that for <v2>, which is consistent with the result from the previous section. However, the simulations with Ms ~ 5 show similar values of α between the decay of <v2> and that of σρ/ρ0 .

Unlike in the case of solenoidal turbulence, γ slightly affects the decay of compressively driven turbulence. First, <v2> in compressively driven turbulence shows bump-like features (indicated by the black arrow in each panel of Figure 3). This slight increase of <v2> is most pronounced in the case Ms ~ 1. Second, dip-like features (indicated by the black arrow in each panel of Figure 4) are visible in the evolution of σρ/ρ0 at about the same time when the bump-like features in <v2> occur. Third, we can see from Figure 4 that the decay of σρ/ρ0 for γ = 0.7 occurs earlier than for γ > 0.7 regardless of Ms.

### 3.2. Decay of Solenoidal and Compressive Velocity Components in Turbulence Driven by Compressive and Solenoidal Drivings

Here, we deal with the differences in the decay of solenoidal and compressive modes. We decompose 3D velocity fields of both solenoidally and compressively driven turbulences into solenoidal and compressive components.3 Figure 5 illustrates the decay of the compressive ratio $〈{\upsilon }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{2}〉/〈{\upsilon }_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2}〉$ (upper panels) and the solenoidal ratio $〈{\upsilon }_{\mathrm{s}\mathrm{o}\mathrm{l}}^{2}〉/〈{\upsilon }_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2}〉$ (bottom panels) in compressively driven turbulence. Here, ${\upsilon }_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2}={\upsilon }_{\mathrm{s}\mathrm{o}\mathrm{l}}^{2}+{\upsilon }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{2}$, and vsol and vcomp are the solenoidal and compressive velocity components, respectively. Blue, red, cyan, and green curves indicate values of γ of 0.7, 1.0, 1.5, and 5/3, respectively.

Decay of the ratio of compressive (upper panels) and solenoidal (bottom panels) velocity components in compressively driven turbulence. Left panels: Ms ~ 1. Center panels: Ms ~ 3. Right panels: Ms ~ 5. Blue, red, cyan, and green curves denote a γ of 0.7, 1.0, 1.5, and 5/3, respectively.

First, Figure 5 shows that the compressive ratio decreases as turbulence decays. When Ms ~ 1 or ~ 3, the smaller γ, the larger the compressive ratio. However, when Ms ~ 5, we do not see a dependence of the compressive ratio on γ. This may be because as Ms increases, gas pressure becomes relatively less important than turbulent pressure and so does the role of the polytropic EOS. Second, and more importantly, the solenoidal ratio increases as turbulence decays and eventually becomes higher than the compressive ratio irrespective of γ and Ms, which means that the compressive velocity component decays faster. Figure 6 shows this for the case of isothermal turbulence initially driven by compressive driving, for which we plot the decay of solenoidal and compressive velocity components. As we can see from the reference lines (dotted lines in different colors in each panel), $〈{\upsilon }_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}^{2}〉$ (magenta curves) decays more quickly than $〈{\upsilon }_{\mathrm{s}\mathrm{o}\mathrm{l}}^{2}〉$ (cyan curves), resulting in larger solenoidal velocity components at the late stages of decay irrespective of Ms.

Decay of solenoidal and compressive velocity components in compressively driven turbulence. Only the case of isothermal turbulence (i.e., γ = 1.0) is presented here. Left: Ms ~ 1. Center: Ms ~ 3. Right: Ms ~ 5. Black, cyan, and magenta curves indicate total, solenoidal, and compressive velocity components, respectively. The dotted lines in different colors are reference lines for different power-law forms. Note that the compressive velocity component decays much faster than the solenoidal one.

While the compressive ratio in compressively driven turbulence becomes less than the solenoidal ratio as the turbulence decays, Figure 7 shows that the compressive ratio in solenoidal turbulence does not. In the figure, blue, red, cyan, and green curves indicate γ of 0.7, 1.0, 1.5, and 5/3, respectively. As we can see from Figure 7, the compressive ratio in solenoidal turbulence is always less than 0.5 regardless of both Ms and polyrtropic γ, implying that the solenoidal velocity component in such turbulence is always dominant over the compressive one, even at the late stages of decay.

Same as Figure 5 but for solenoidally driven turbulence and showing the compressive ratio only.

### 3.3. Density PDF and Skewness

In the following, we investigate the density PDF and its skewness for the case of compressively driven turbulence with polytropic EOS in driven and decay regimes. We define the skewness of the density PDF as

 $\mathrm{S}\mathrm{k}\mathrm{e}\mathrm{w}\left(s\right)=\frac{1}{N}\sum _{i=1}^{N}{\left(\frac{{s}_{i}-〈s〉}{{\sigma }_{s}}\right)}^{3}$ (6)

where N is the total number of data points, s ≡ ln(ρ/ρ0) is the natural logarithm of the density fluctuations, and <· · ·> denotes the spatial average. Skewness measures the asymmetry of a probability distribution. When the distribution is left-skewed (right-skewed), the skewness takes a negative (positive) value.

Figures 8 and 9 show the density PDFs of polytropic turbulence with Ms ~ 1 and ~ 3, respectively. Each colored solid curve corresponds to the density PDF at a different time along the decay. We carry out log-normal fitting for the PDFs using the relation

 ${p}_{s}\left(s\right)=\frac{1}{\sqrt{2\pi {\sigma }_{s}}}\mathrm{exp}\left[-\frac{{\left(s-〈s〉\right)}^{2}}{2{\sigma }_{s}^{2}}\right].$ (7)

PDF of the logarithmic density s = ln(ρ/ρ0) of decaying polytropic turbulence initially driven by compressive driving with Ms ~ 1. Left: polytropic γ = 0.7. Center: γ = 1.0. Right: γ = 5/3. Note that the p(s) axis is logarithmic. Black, purple, and orange curves represent different times along the decay. Dotted lines are log-normal fitting line.

Same as Figure 8, but for Ms ~ 3. Left: γ = 0.7. Center: γ = 1.0. Right: γ = 1.5.

In the figures, fitting lines are indicated by dotted lines.

We first consider driven turbulence (black solid line in Figures 8 and 9). Compressively driven turbulence yields density PDFs which are not exactly log-normal even in the case γ = 1. The density PDF for γ = 0.7 is slightly right-skewed, and that for γ = 5/3 is strongly left-skewed. The latter has a pronounced power-law tail at low density as shown in Figure 8c. Figure 9c shows that the PDF of CMS3-γ1.5, which is for Ms ~ 3, deviates more strongly from the log-normal form at low density. However, it is not clear whether the low density tail follows a power-law.

We can interpret the behavior of density PDF with γ stated above as follows. When γ < 1 (γ > 1), compression decreases (increases) internal temperature and thus local sound speed, which allows (hinders) high density regions to develop easily (Federrath & Banerjee 2015). Since compression can commonly occur in our simulations of transonic/supersonic turbulence with compressive driving, the density PDFs with γ < 1 (γ > 1) become right-skewed (left-skewed). In addition, although density PDFs with γ = 0.7 are slightly right-skewed, we expect that they become more right-skewed as γ decreases further. In fact, Scalo et al. (1998) showed a clear power-law tail toward high density regions in their one-dimensional simulation with γ = 0.3 and Ms ~ 3.

Note that our results are for compressively driven turbulence with polytropic EOS. Earlier studies are available for solenoidal turbulence with polytropic EOS and Ms ~ 10 (Federrath & Banerjee 2015), and for compressively driven turbulence with γ = 1 and Ms ~ 5 (Federrath et al. 2010). Our current result is consistent with that of the latter reference in that the PDF is slightly left-skewed when γ = 1. Our result is also consistent with that of the former reference in that the PDF is strongly left-skewed with a power-law tail when γ = 5/3. However, the PDF of solenoidal turbulence is more or less symmetric when γ = 0.7 or 1.0 (Federrath & Banerjee 2015), while that of compressively driven turbulence is clearly right-skewed for transonic turbulence and slightly right-skewed in supersonic turbulence when γ = 0.7 (see Figures 8a and 9a).

Figure 10 shows the time evolution of the skewness of the density PDF for compressively driven turbulence. The horizontal axis denotes the elapsed decay time normalized by the large-eddy turnover time. The top and bottom panels of Figure 10 show the skewness for Ms ~ 1 and ~ 3, respectively. At t = 0, the PDF for γ > 1 has a negative skewness (see green curve in the top panel and cyan curve in the bottom panel). Likewise, the skewness for other values of γ indicates that the density PDFs presented in the figure deviate from a log-normal form at t = 0.

Time evolution of the skewness of the density PDFs shown in Figure 8 (top panel) and Figure 9 (bottom panel). Blue, red, cyan, and green curves correspond to γ of 0.7, 1.0, 1.5, and 5/3, respectively.

Skewness of the PDFs presented in Figures 8 and 9

Next, we consider the decay regime. As can be seen from the purple and orange solid curves in Figures 8 and 9, as turbulence decays, the density PDFs become narrow and approach log-normal forms in all cases. We can confirm this trend in Figure 10. The skewness for Ms ~ 1 and ~ 3 approaches, and fluctuates around, zero as turbulence decays, which is consistent with the temporal change of the PDFs shown in Figures 8 and 9.

## 4. DISCUSSION AND SUMMARY

The purpose of this study is to investigate the effects of the equation of state (i.e., the value of the polytropic index γ) and driving schemes (i.e., solenoidal and compressive drivings) on decaying turbulence and its statistics. In this paper, we demonstrated that the scaling relation for the decay law (<v2> ∝ tα) does not show a strong dependence on γ and the driving schemes. Throughout our simulations, <v2> decays with 0.8 ≾ α ≾ 1.2. The range is nearly same as the one found by Mac Low et al. (1998) (0.85 < α < 1.2).

For γ > 1 cases, α ranges from 1.0 to 1.2 in solenoidal turbulence and from 0.8 to 1.0 in compressively driven turbulence, with the largest value of α being obtained for the case Ms ~ 1 in both driving schemes. This result confirms the assumption of Davidovits & Fisch (2017) that for γ = 5/3, α falls into the range 1.0–1.5 with a weak dependence on the initial Mach number.

Even though no relationship between polytropic index γ and scaling of the decay law (<v2> ∝ tα) is found in our study, there are several noticeable characteristics in case of compressively driven turbulence. First, a slight increase of <v2> and an associated decrease of σρ/ρ0 are found in Figures 3 and 4, respectively. As discussed earlier, those effects can be interpreted as being due to additional energy released from compressed regions via expansion. Second, in the case Ms ~ 1, the effect is most pronounced. This may be due to the pressure being strong compared to that of supersonic cases, resulting in stronger expansion. Third, for the same Ms, bump and dip like features are more prominent for γ = 0.7. This is because, for compressive driving, turbulent gas with γ = 0.7 is more easily compressed. Thus, expansion is easier when turbulence decays, leading to the stronger feature for the case γ = 0.7. Lastly, σρ/ρ0 decays more quickly for γ = 0.7 than for γ > 0.7 in both transonic and supersonic turbulence driven by compressive driving. This is beuase when γ is less than one, expansion increases the internal temperature, which dissipates density structures quickly. Therefore, the density decays faster for smaller γ.

More interestingly, the compressive velocity component decays faster than the solenoidal one in the case of turbulence initially driven by compressive driving as shown in Section 3.2. We can interpret this as follows. When turbulence initially driven by compressive driving decays, the energy of the compressive component is dissipated through both turbulent cascades and dissipation at shocks, and a fraction of the energy is converted into solenoidal energy. For the solenoidal component, only turbulent cascades can dissipate the energy. As it has fewer channels for energy dissipation available than the compressive component, the solenoidal velocity component in turbulence initially driven by compressive driving decays more slowly. However, a detailed analysis addressing, e.g., what fraction of compressive kinetic energy changes into solenoidal kinetic energy, is beyond the scope of this paper; further studies are required to understand this issue quantitatively.

Our study uses polytropic EOS to quantify decaying astrophysical turbulence. A fluid with polytropic EOS (P = γ) is different from an ideal gas with adiabatic index γ > 1 as follows. First, we do not need to solve the energy conservation equation for the former, while we have to for the latter. Note that the EOS for the latter becomes P = (γ − 1)E, where E is the internal energy density. Second, they provide different descriptions of shocks. The polytropic EOS is only applicable to isentropic flows; the entropy is constant everywhere, even in shocked regions. The entropy is not constant for an ideal gas. Although entropy is conserved in regions with adiabatic expansion/compression, it increases when a parcel of gas passes through a shock front in an ideal gas. Therefore, a fluid with polytropic EOS and an ideal gas with adiabatic index γ have different descriptions of shocks and their shock jump conditions are different (see Bisnovatyi-Kogan & Moiseenko 2016 for shock jump conditions for adiabatic and isentropic flows).

Our results have astrophysical implications. The polytropic exponent γ is useful to describe a variety of components in the ISM. A polytropic EOS with γ ⋍ 0.8 can represent the density range of 10cm−3n ≤ 104cm−3 , where n is the hydrogen number density (Glover & Mac Low 2007b); this corresponds to the density range for giant molecular clouds (Ferrière 2001). Furthermore, the EOS with γ ~ 1.4 describes the conditions in the centers of protostellar cores, with a corresponding density range 1012cm−3n ≤ 1017cm−3 (Masunaga & Inutsuka 2000). Accordingly, our results suggest that once driving of turbulence ceases to act, turbulence quickly decays with a dynamical timescale that is characteristic of a certain system irrespective of its spatial scale. Moreover, even if turbulence is initially driven by compressive driving like supernova explosions, solenoidal motions will dominate as the turbulence decays due to the much faster decay of compressive motions.

In summary, we have studied the influence of polytropic EOS and driving schemes on decaying turbulence and its statistics and found the following results.

• 1. There is no significant correlation between the scaling of the decay law (Etα) and polytropic exponent γ in the case of solenodially driven turbulence.
• 2. Driving schemes have a non-negligible effect on the decay rate of turbulence: the power-law index α for turbulence initially driven by compressive driving is smaller than that for turbulence initially driven by solenoidal driving.
• 3. There is no significant effect of γ on the decay rate of velocity in compressively driven turbulence.
• 4. The polytropic exponent γ has a small effect on the density fluctuations in compressively driven turbulence: the smaller γ is, the faster the standard deviation of density fluctuations of the turbulence decays.
• 5. When considering the decay of solenoidal and compressive velocity components in compressively driven turbulence separately, the energy of the compressive velocity component decays much faster.
• 6. Regarding the statistics of compressively driven turbulence, the density PDF deviates from a log-normal distribution, especially for γ > 1. In addition, we have found that the skewness of the density PDF of the turbulence becomes zero as it decays.

## Acknowledgments

This paper has been expanded from a chapter of Jeonghoon Lims MSc thesis. This work is supported by the National R&D Program through the National Research Foundation of Korea Grants funded by the Korean Government (NRF-2016R1A5A1013277 and NRF-2016R1D1A1B02015014).

Notes
1We are interested in the decay of <(δv)2>, where δv is the magnitude of the velocity fluctuations. If <v> is non-zero, <(δv)2> is different from <v2>. That is, <v2> = <(δv)2>+<vx>2 +<vy>2 +<vz>2 , where <vi> (i = x, y, z) is the mean velocity component over the whole computational box. In Runs SMS1-γ1.0, SMS3-γ1.0, and SMS5-γ1.0, <vx>, <vy>, and <vz> are less than ~0.01 at the moment turbulence starts decaying. Therefore, we can ignore the effect of non-zero <vi>. We assume that this holds true also for the other simulations presented in this paper.
2By “solenoidal turbulence” we mean turbulence initially driven by solenoidal driving.
3To be specific, we carry out Fourier transforms of total velocity fields first. Then, we take only the velocity components perpendicular to k, which correspond to solenoidal velocity component in Fourier space, and carry out inverse Fourier transforms to produce solenoidal velocity fields in real space. Finally, we subtract the solenoidal velocity components from the total velocities in real space to yield the compressive velocity components.

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### Figure 1.

Time evolution of <v2> in decaying turbulence initially driven by solenoidal driving with γ = 0.7 (blue), 1.0 (red), 1.5 (cyan), or 5/3 (green), where <· · ·> indicates spatial averaging. Left: Ms ~ 1. Center: Ms ~ 3. Right: Ms ~ 5. Turbulence starts decaying at tcode/ted = 0. We normalize <v2> by υ02 which is the value at tcode/ted = 0. The black dotted lines in all three panels are reference lines for different power-law exponents.

### Figure 2.

Same as Figure 1, but for the decay of the standard deviation of the density fluctuation σρ/ρ0 . We normalize σρ/ρ0 by (σρ/ρ0)0 which is the value at tcode/ted = 0.

### Figure 3.

Same as Figure 1, but for compressively driven turbulence. Note that unlike Figure 1, <v2> slightly increases along the decay as indicated by a black arrow in each panel; this is most apparent in the case of Ms ~ 1 (left panel).

### Figure 4.

Same as Figure 2, but for compressively driven turbulence.

### Figure 5.

Decay of the ratio of compressive (upper panels) and solenoidal (bottom panels) velocity components in compressively driven turbulence. Left panels: Ms ~ 1. Center panels: Ms ~ 3. Right panels: Ms ~ 5. Blue, red, cyan, and green curves denote a γ of 0.7, 1.0, 1.5, and 5/3, respectively.

### Figure 6.

Decay of solenoidal and compressive velocity components in compressively driven turbulence. Only the case of isothermal turbulence (i.e., γ = 1.0) is presented here. Left: Ms ~ 1. Center: Ms ~ 3. Right: Ms ~ 5. Black, cyan, and magenta curves indicate total, solenoidal, and compressive velocity components, respectively. The dotted lines in different colors are reference lines for different power-law forms. Note that the compressive velocity component decays much faster than the solenoidal one.

### Figure 7.

Same as Figure 5 but for solenoidally driven turbulence and showing the compressive ratio only.

### Figure 8.

PDF of the logarithmic density s = ln(ρ/ρ0) of decaying polytropic turbulence initially driven by compressive driving with Ms ~ 1. Left: polytropic γ = 0.7. Center: γ = 1.0. Right: γ = 5/3. Note that the p(s) axis is logarithmic. Black, purple, and orange curves represent different times along the decay. Dotted lines are log-normal fitting line.

### Figure 9.

Same as Figure 8, but for Ms ~ 3. Left: γ = 0.7. Center: γ = 1.0. Right: γ = 1.5.

### Figure 10.

Time evolution of the skewness of the density PDFs shown in Figure 8 (top panel) and Figure 9 (bottom panel). Blue, red, cyan, and green curves correspond to γ of 0.7, 1.0, 1.5, and 5/3, respectively.

### Table 1

Simulation runs and parameter values

Run Drivinga γb Resolution kcf Mds αe1 αf2 (t1, t2)g
We did not include a stiff EOS simulation (i.e., γ > 1) for compressively driven turbulence with Ms ≿ 3 and for solenoidally driven turbulence with Ms ≿ 5. This is because the simulations crashed before turbulence saturates due to the high Ms and large γ, which can lead to extremely low density values (see the PDFs in Figures 8 and 9 for γ > 1). Therefore, we could not run CMS3-γ5/3, CMS5-γ5/3, or SMS5-γ5/3. When γ = 5/3 simulations are unstable, we use γ = 1.5 instead for SMS5 and CMS3. Using different γ for the stiff EOS would not be appropriate for a comparison. Nevertheless, we think that the difference between γ = 1.5 and 5/3 would not significantly affect the decay law. Even γ = 1.5 is unstable for CMS5. Therefore, we do not have data for CMS5 with a stiff EOS.
a Driving schemes – either solenoidal or compressive driving.
b Polytropic exponent.
c The driving wavenumber at which the energy injection rate peaks.
d The sonic Mach number which is defined in Equation (5).
e Average of power law exponent for v2 ∝ t−α1 .
f Average of power law exponent for σρ/ρ0 ∝ t−α2 .
g Time interval in units of large-eddy turnover time (ted) for averaging α1 and α2.
SMS1-γ0.7 Solenoidal 0.7 5123 8.0 ~1 1.185 0.279 (5,10)
SMS1-γ1.0 Solenoidal 1.0 5123 8.0 ~1 1.202 0.291 (5,10)
SMS1-γ5/3 Solenoidal 5/3 5123 8.0 ~1 1.313 0.289 (5,10)
SMS3-γ0.7 Solenoidal 0.7 5123 8.0 ~3 1.118 0.425 (5,10)
SMS3-γ1.0 Solenoidal 1.0 5123 8.0 ~3 1.140 0.469 (5,10)
SMS3-γ5/3 Solenoidal 5/3 5123 8.0 ~3 1.166 0.567 (5,10)
SMS5-γ0.7 Solenoidal 0.7 5123 8.0 ~5 0.971 0.644 (5,10)
SMS5-γ1.0 Solenoidal 1.0 5123 8.0 ~5 0.949 0.425 (5,10)
SMS5-γ1.5 Solenoidal 1.5 5123 8.0 ~5 1.073 0.650 (5,10)
CMS1-γ0.7 Compressive 0.7 5123 8.0 ~1 0.954 0.520 (5,10)
CMS1-γ1.0 Compressive 1.0 5123 8.0 ~1 0.825 0.599 (5,10)
CMS1-γ5/3 Compressive 5/3 5123 8.0 ~1 1.058 0.500 (5,10)
CMS3-γ0.7 Compressive 0.7 5123 8.0 ~3 0.777 0.522 (5,10)
CMS3-γ1.0 Compressive 1.0 5123 8.0 ~3 0.818 0.481 (5,10)
CMS3-γ1.5 Compressive 1.5 5123 6.0 ~3 0.839 0.461 (5,10)
CMS5-γ0.7 Compressive 0.7 5123 8.0 ~5 0.688 0.606 (5,10)
CMS5-γ1.0 Compressive 1.0 5123 8.0 ~5 0.767 0.513 (5,10)

### Table 2

Skewness of the PDFs presented in Figures 8 and 9

Run ta1 tb2 tc3
a Time marked by black solid lines in Figures 8 and 9.
b Time marked by purple solid lines in Figures 8 and 9.
c Time marked by orange solid lines in Figures 8 and 9.
CMS1-γ0.7
CMS1-γ1.0
CMS1-γ5/3
CMS3-γ0.7
CMS3-γ1.0
CMS3-γ1.5
0.260
−0.335
−1.576
0.204
−0.301
−0.784
0.084
0.290
0.168
0.197
−0.149
−0.382
0.294
0.260
0.260
0.093
0.070
−0.013