turbulent heating of galaxy-cluster plasmas - IOPscience

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Timothy J. Dennis and Benjamin D. G. Chandran. Department of Physics and Astronomy, University of Iowa, 203 Van Allen Hall, Iowa City, IA 52242;.
The Astrophysical Journal, 622:205 –216, 2005 March 20 # 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.

TURBULENT HEATING OF GALAXY-CLUSTER PLASMAS Timothy J. Dennis and Benjamin D. G. Chandran Department of Physics and Astronomy, University of Iowa, 203 Van Allen Hall, Iowa City, IA 52242; [email protected], [email protected] Receivved 2003 October 3; accepted 2004 November 6

ABSTRACT A number of studies suggest that turbulent heating plays an important role in the thermal balance of galaxycluster plasmas. In this paper, we construct a model of intracluster plasmas in which radiative cooling is balanced by heating from viscous dissipation of turbulent motions, turbulent diffusion of high–specific-entropy plasma into low–specific-entropy regions, and thermal conduction. We solve for the rms turbulent velocity u by setting þ Q þ H ¼ R throughout a cluster, where , Q, and H are the heating rates from dissipation of turbulence, turbulent diffusion, and conduction, respectively, and R is the rate of radiative cooling. We account for the effects of buoyancy in our expression for the eddy diffusivity and neglect nonthermal pressure. We take the conductivity to be a fixed fraction (typically one-fifth) of the Spitzer value for a nonmagnetized plasma and the density and temperature to be given by analytical fits to published data. We set the dominant velocity length scale l equal to  r þ l0 , where  is a constant, r is distance from cluster center, and l0 ¼ 0:5 kpc. For 0:05 <  < 1, we find velocities in the range 100 P u P 300 km s1. The inclusion of dissipation substantially reduces the value of u needed to balance cooling when  P 0:5, relative to models in which turbulent diffusion is the only form of turbulent heating. We find that k Q when  < 0:5, and P Q when  > 0:5, although there are exceptions to this rule. For some values of , we find that at some locations the heat flux from turbulent diffusion has positive divergence, so that turbulent diffusion locally cools the plasma. Buoyancy inhibits turbulent diffusion of heat in the radial direction to a degree that increases with increasing  . This leads to an increase in the computed value of u relative to models that neglect buoyancy; the magnitude of the increase is moderate for  ¼ 0:5 and large for  > 1. Subject headinggs: galaxies: clusters: general — galaxies: clusters: individual (A1795, A2199, 3C 295, RX J1347.51145) — turbulence

1. INTRODUCTION

Ciotti & Ostriker 2001; Churazov et al. 2002; Bru¨ggen et al. 2002; Ruszkowski & Begelman 2002; Fabian et al. 2003; Reynolds et al. 2005; Chandran 2004), thermal conduction ( Binney & Cowie 1981; Chandran & Cowley 1998; Narayan & Medvedev 2001; Gruzinov 2002; Voigt et al. 2002; Zakamska & Narayan 2003; Kim & Narayan 2003a; Chandran & Maron 2004; Maron et al. 2004), dissipation of turbulent motions (Loewenstein & Fabian 1990; Churazov et al. 2004), and turbulent diffusion of heat (Cho et al. 2003; Kim & Narayan 2003b; Voigt & Fabian 2004). However, the solution to the coolingflow problem remains unclear. In the heated cooling-flow paradigm, the heating mechanism(s) must satisfy a number of requirements. First, the heat source(s) must be capable of offsetting radiative cooling for average cluster temperatures ranging from 2 keV in less massive clusters to 15 keV in the most massive clusters. Second, the heating mechanism must lead to equilibrium density and temperature profiles that are consistent with observations. Third, the balance between heating and cooling must be self-regulating and must not require artificial fine-tuning. This paper focuses on turbulent heating. The importance of turbulent heating is suggested by recent observations by Churazov et al. (2004), who found evidence for plasma motion in the Perseus cluster at roughly half the sound speed. Kim & Narayan (2003b) and Voigt & Fabian (2004) investigated heated cooling-flow models in which radiative cooling is balanced by turbulent diffusion of heat toward cluster center from the relatively hotter and higher entropy plasma surrounding a cluster’s core. They showed that turbulent diffusion can satisfy the first and second of the above requirements if the turbulent velocity and dominant turbulence length scale have the right

Galaxy clusters are filled with vast quantities of diffuse, hot (107 –108 K) plasma, and the radiative cooling time at the centers of many clusters is much shorter than a typical cluster’s age (Fabian 1994). In the cooling-flow model (Fabian 1994), heating of intracluster plasma is neglected, and a steady state is assumed. Radiative cooling and gravity then cause a subsonic ˙ CF that exceeds 103 M yr1 in some inflow and a cooling rate M clusters. The cooling-flow model also predicts copious line emission from ionization stages such as Fe xvii that arise at temperatures between 106 and 107 K. However, recent high spectral resolution X-ray observations fail to detect this emission, con˙ CF (Peterson straining the cooling rate in clusters to be P 0.1M et al. 2001; Tamura et al. 2001; Fabian 2003). The challenge of explaining this discrepancy is the so-called cooling-flow problem. Different types of solutions to the cooling-flow problem have been proposed. For example, there is the ‘‘heating-flow’’ paradigm, in which gas or plasma is fed into the central region (from, e.g., stellar winds), and some heating mechanism overpowers radiative cooling, leading to an outflow from the central region (Takahara & Takahara 1979). There are also many versions of the ‘‘heated cooling-flow’’ model, in which radiative cooling is partially or completely balanced by some source of heating. A number of heating mechanisms have been proposed, including galaxy motions (Bregman & David 1989; El-Zant et al. 2004), supernovae (Bregman & David 1989), cosmic rays ( Bohringer & Morfill 1988; Loewenstein et al. 1991; Tucker & Rosner 1983), active galactic nuclei ( Rosner & Tucker 1989; Pedlar et al. 1990; Tabor & Binney 1993; Binney & Tabor 1995; 205

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Vol. 622

TABLE 1 Parameters for the Plasma Density, Temperature, and NFW Dark Matter Density Profile

Cluster

T0 ( keV)

T1 ( keV)

rct (kpc)



ne(0) (cm3)

rc1 (kpc)

rc2 (kpc)



2

M0 (1014 M)

rs ( kpc)

A2597....... A1795....... A478.........

7.74 7.01 8.72

6.5 4.75 7.00

9.5 47.2 27.6

0.0790 0.576 0.487

0.0982 0.0540 0.152

2.6 169 384

42.6 18.2 9.52

0.0967 0.2740 0.6690

0.816 0.451 0.505

2.800 6.600 6.528

360 460 492

values. However, it is not yet clear whether turbulent heating satisfies the third requirement. In particular, it is not clear how intracluster turbulence is generated or how the turbulence amplitude attains the value needed for turbulent heating to balance cooling. In this paper, we further explore turbulent heating in intracluster plasmas and develop a heated cooling-flow model in which radiative cooling is balanced by a combination of turbulent diffusion, viscous (or collisionless) dissipation of turbulent motions, and thermal conduction. Our goals are (1) to determine how the inclusion of dissipation of turbulence affects the required turbulent velocities, (2) to compare the importance of turbulent diffusion and dissipation of turbulence, (3) to assess the importance of buoyancy forces on turbulent heating, and (4) to shed further light on the seriousness of the fine-tuning problem for turbulent heating models. We start by fitting density and temperature data for three different clusters (A478, A1795, and A2597) with analytic profiles (the parameters are given in Table 1). We then solve for the rms turbulent velocity u as a function of radius r by setting þ Q þ H ¼ R, where , Q, and H are the heating rates from dissipation of turbulence, turbulent diffusion, and thermal conduction, respectively, and R is the rate of radiative cooling. We take the conductivity to be a fixed fraction f (typically 0.2) of the Spitzer value for a nonmagnetized plasma (Spitzer 1962) and set the dominant velocity length scale l equal to r þ l0 , where  is a free-parameter constant, and l0 ¼ 0:5 kpc. Because the specific entropy of intracluster plasma increases outward, the plasma is convectively stable if nonthermal pressure is ignored. If the nonlinear turbulent frequency u=l is less than the Brunt-Vaisala frequency NBV of buoyancy oscillations, then radial motions are primarily oscillatory and only weakly turbulent, and turbulent diffusion is less efficient than in strong turbulence. We take this into account in our expression for Q. For 0:05 <  < 1, we find velocities in the range 100 P u P 300 km s1. Roughly speaking, we find that k Q when  < 0:5, and P Q when  > 0:5, although there are exceptions to this rule. In general, both dissipation and diffusion are important. The inclusion of dissipation substantially reduces the value of u needed to balance cooling when  P 0:5, relative to models in which turbulent diffusion is the only form of turbulent heating. For some values of  and at some locations, the heat flux from turbulent diffusion can have positive divergence, so that turbulent diffusion locally cools the plasma. The inclusion of buoyancy effects leads to an increase in the computed value of u, relative to models in which buoyancy effects are neglected. The magnitude of this increase is moderate for  ¼ 0:5 and large for  > 1. Because the heating rate from dissipation of turbulence satisfies ’ u3 =l, where  is the plasma density, moderate changes in u lead to substantial changes in . As u is varied from 0.1cs to cs (where cs is the sound speed), the ratio =R varies from a value much less than 1 to a value much greater than 1. This finding emphasizes the point made above that if turbulent heating

solves the cooling-flow problem by approximately balancing radiative cooling, an explanation is needed for the fine-tuning of u to the required value. Since turbulent heating is simply a vehicle for transferring power to diffuse intracluster plasma from the energy source that generates the turbulence, a solution to the fine-tuning problem will require an identification of the underlying energy source. Although such a solution is beyond the scope of this paper, we mention two possibilities that have been discussed in the literature. El-Zant et al. (2004) considered plasma heating from galaxy motions and argued that such a mechanism is selfregulating in that the heating is efficient only when the plasma sound speed is smaller than the galaxy velocity dispersion . They suggest that plasma cools only to the point that the sound speed drops moderately below , at which point efficient heating prevents further cooling. Although they do not discuss turbulent heating explicitly, the connection between their work and turbulent heating could be made. A second possibility is that the turbulence is generated by a central active galactic nucleus (AGN) (Churazov et al. 2004; Chandran 2004). One of the ways in which a central AGN can generate plasma turbulence is by producing cosmic rays that mix into the thermal intracluster plasma. If the cosmic-ray pressure gradient is sufficiently large, cosmic-ray buoyancy drives convection in the thermal plasma, although the specific entropy of the thermal plasma increases outward (Chandran 2004). Further work on the excitation of intracluster turbulence will be very important for understanding the role of turbulent heating in intracluster plasmas. In x 2 we review results in the literature on the heating rates associated with dissipation of turbulent motions and turbulent diffusion of heat. In x 3 we present the equations of our heated cooling-flow model. In x 4 we discuss the results of our model equations for different choices of model parameters, and we summarize our findings in x 5. 2. THE HEATING RATES FOR DISSIPATION OF TURBULENCE AND TURBULENT DIFFUSION OF HEAT The rate of heating per unit volume from dissipation of kinetic and magnetic energy in strong magnetohydrodynamic (MHD) turbulence is given by1 ¼

cdiss u3 ; l

ð1Þ

where  is the mass density of the fluid, cdiss is a dimensionless constant, and l is the dominant velocity length scale. The value 1 If the fluctuating velocity and magnetic fields become aligned in decaying high- MHD turbulence (i.e., large fractional cross-helicity), diss can be reduced significantly relative to eq. (1) ( Dobrowolny et al. 1980; Grappin et al. 1983). In clusters, however, such preferential alignment is not expected (see, e.g., Chandran & Rodriguez 1997).

No. 1, 2005

TURBULENT HEATING OF GALAXY-CLUSTER PLASMAS

of l is not well known for intracluster turbulence. In our heated cooling-flow model, described in detail in x 3, we set l ¼  r þ l0 ;

ð2Þ

where  is an adjustable constant, and l0 ¼ 0:5 kpc, so that l increases with radius r. However, we determine cdiss using results from the literature on periodic homogenous turbulence in which l does not vary in space. To do this, we employ the following operational definition of l for periodic turbulence, l¼

 ; kp

ð3Þ

where kp is the maximum of kE(k), and E(k) is the kinetic energy spectrum [the mean-square velocity being u2 ¼ R1 2 0 E(k) dk]. Thus, kp is the wavenumber of the fluctuations that make the dominant contribution to the total kinetic energy. We assign cdiss the value that is obtained in simulation E of Haugen et al. (2004), a 10243 periodic simulation of forced, compressible, subsonic MHD turbulence with no helicity and no mean magnetic field. In this simulation, ¼ 0:135kp u3 , which yields cdiss ¼ 0:42:

ð4Þ

The integral length scale lI is an alternative measure of the length scale of the dominant large-scale eddies in a turbulent flow. The relation between lI and l depends on the form of E(k) at small wavenumbers. In the Appendix, it is shown that lI ¼ 9l=20 for a simple model spectrum. In the 40963 simulations of incompressible hydrodynamic turbulence of Kaneda et al. (2003), the rate of energy dissipation per unit mass is   ð5Þ hydro ¼ 0:41 u3x =lI ; where u2x ¼ u2 =3. Using the conversion lI ¼ 9l=20 and converting to a rate of energy dissipation per unit volume, hydro, the result of Kaneda et al. (2003) becomes hydro ¼ 0:18u3 =l, roughly half the dissipation rate in MHD turbulence with the same u and l. The heating per unit volume from turbulent diffusion is denoted Q and is set equal to Q ¼ : = (Deddy T :s);

ð6Þ

where s ¼ CV ln ( p= ) is the specific entropy, p is the pressure, T is the temperature,  ¼ Cp =CV ¼ 5=3 is the ratio of specific heats, CV ¼ 3kB =2 mH , ¼ =nmH is the mean molecular weight, and Deddy is the eddy diffusivity. There are two subtleties for the evaluation of Deddy. First, clusters are filled with chaotic magnetic fields (Kronberg 1994; Taylor et al. 2001, 2002). The magnetic field strengths (typically several G) inferred from Faraday rotation suggest magnetic energies that are comparable to the kinetic energies of the turbulent velocities in our model. There is some disagreement in the literature over the efficiency of turbulent diffusion in MHD turbulence. Some authors have argued that the eddy diffusivity Deddy is Tul, since magnetic tension inhibits the wandering of fluid parcels (Vainshtein & Rosner 1991; Cattaneo 1994), while others have argued that Deddy  ul as in hydrodynamic turbulence (Cho et al. 2003). In this paper, we neglect the effects of magnetic tension but note that this may cause us to overestimate the role of turbulent diffusion in clusters.

207

The second subtlety concerns the role of buoyancy. The specific entropy is observed to increase outward in clusters. In this paper, we neglect nonthermal pressure from magnetic fields and cosmic rays, so that the intracluster medium is convectively stable and supports radial buoyancy oscillations at the BruntVaisala frequency,

NBV

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 dp 1 d  ¼ g ; p dr  dr

ð7Þ

where g is the gravitational acceleration. In the absence of buoyancy, the dominant turbulent eddies of scale l are randomized after one eddy-turnover time, l=u, and a fluid element undergoes a random walk of step length x ’ l and time step t ’ l=u, leading to a diffusion coefficient Deddy  ( x)2 =t  ul. However, if the dominant eddy-turnover frequency u=l isTNBV, then the radial motions are primarily oscillatory with frequency NBV, and the turbulence is described as ‘‘weak.’’ Over the course of several oscillation periods, the radial displacement of a fluid element oscillates between  r, where r  u=NBV , a value that is Tl. If one were to adopt a simplistic view in which the flow is abruptly randomized after a time t  l=u (randomization on this timescale being provided by strongly turbulent motions perpendicular to the rˆ -direction), then the radial diffusion coefficient becomes Dr  (r)2 =t  ul(u=lNBV )2 . Thus, when u=lTNBV , buoyancy dramatically suppresses radial diffusion of heat. We account for the effects of buoyancy by employing the results of Weinstock (1981), Deddy ¼ ctd ul ;

ð8Þ

1 ; 1 þ c 20 l 2 N 2BV =u2

ð9Þ

where

¼

c20 ¼ 0:1688, and ctd is a dimensionless constant. Equation (8) interpolates between the weak-turbulence (u=lTNBV ) and strong-turbulence (u=l 3 NBV ) limits. We evaluate ctd by demanding that the strong-turbulence limit ( ! 1) of equation (8) reproduce the eddy diffusivity for passive scalar diffusion in incompressible homogeneous hydrodynamic turbulence. It follows from, for example, equations (106) and (112) of Chandrasekhar (1943) and equation (9.35) of Monin & Yaglom (1965) that Deddy ¼ u2 TL =3, where TL is the Lagrangian integral timescale. Using direct numerical simulations of hydrodynamic turbulence, Yeung (1994) finds TL ¼ 35lI =46u. Using the conversion lI ¼ 9l=20, Yeung’s (1994) result gives ctd ¼ 0:11:

ð10Þ

This value differs only slightly from Weinstock’s (1981) value (ctd  0:13). For comparison, we list several other values of ctd that can be inferred from the literature. Lesieur (1987) provides 4=3 1=3 the estimate Deddy ¼ 0:3l I hydro for hydrodynamic turbulence. Using equation (5) and the conversion lI ¼ 9l=20, Lesieur’s (1987) estimate yields ctd ¼ 0:06. In K– models, the eddy diffusivity is cK u4 =4. Empirical fits give cK ¼ 0:1, while a directinteraction–approximation analysis gives cK ¼ 0:13 (Shimomura 1998). Using equation (5) and the conversion lI ¼ 9l=20, the value cK ¼ 0:1 yields ctd ¼ 0:14, and the value cK ¼ 0:13 yields ctd ¼ 0:18.

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DENNIS & CHANDRAN

We note that even when u=lTNBV , only the radial component of the motion becomes highly oscillatory, and the motions perpendicular to rˆ remain nonoscillatory and strongly turbulent. We assume that both the radial and nonradial motions are characterized by the same values of u and l. Thus, buoyancy has only a small effect on the dissipation of turbulent motions, which we ignore. We take the gravitational acceleration to be dominated by a cluster’s dark matter, so that g¼

GMr ; r2

ð11Þ

M0 =2 rðr þ rs Þ2

R  n2e 

"



kB T ¼ ni ne 0:0086 1 keV

1:7



kB T þ0:058 1 keV

#

0:5 þ0:063

; 1022 ergs cm3 s1 ;

where Mr is mass of dark matter within a distance r of cluster center. To calculate Mr , we take the dark-matter density distribution  DM(r) to have a Navarro-Frenk-White (NFW) profile (Navarro et al. 1997). In the notation of Zakamska & Narayan (2003), DM (r) ¼

the effects of turbulent cluster magnetic fields, which inhibit the diffusion of individual particles and heat. Recent studies indicate that f  0:1–0.2 (Narayan & Medvedev 2001; Chandran & Maron 2004; Maron et al. 2004). We approximate the radiative cooling function for free-free and line emission with the equation

;

the concentration parameter c is given by  1=3 1 3Mvir c¼ ; rs 4200crit (z)

Equation (19) leads to a differential equation for the rms velocity u, "   du u3 d 2 ¼ R  H  cdiss   u

þ dr dr r l #"   #1  2c20 2 l dNBV 2c 20 l 2 N 2BV 2 þ ; NBV þ lNBV 1 þ u dr u2 ð20Þ

ð15Þ

3. MODEL EQUATIONS We assume that radiative cooling is everywhere balanced by heating from viscous dissipation of turbulence, turbulent diffusion, and thermal conduction. The volumetric heating rate from thermal conduction is given by

¼ ctd lT

ds : dr

ð21Þ

We take  and T to be known functions of r determined by fitting temperature and density data for the clusters A478, A1795, and A2597 ( provided by S. Allen, S. Ettori, and B. McNamara; see Sun et al. 2003; Ettori, et al. 2002; McNamara et al. 2001). For the temperature fits, we use the expression "  2 # r ; ð22Þ T (r) ¼ T0  T1 1 þ rct and for the density fits we use "  2 #3=2 "  2 #2 r r 1þ ; (r) ¼ 0 1 þ rc1 rc2

ð23Þ

ð16Þ

The thermal conductivity is set equal to the Spitzer value (1962) multiplied by a suppression factor f,     kB T 5=2 T ¼ 9:2 ; 1030 ne k B f 5 keV  2   3 10 cm 37 ; cm2 s1 ; ln C ne

ð19Þ

where we have defined

is the critical density of the universe at the redshift of the cluster. For two of the three clusters examined in this paper (A1795 and A2597), Zakamska & Narayan (2003) provide values for the parameters M0 and rs. For the cluster A478, we use the recent observations of Pointecouteau et al. (2004), who provide values of rs and Mvir , and we assume H0 ¼ 70 km s1 Mpc1, M ¼ 0:3, and  ¼ 0:7. The values of M0 and rs for the two clusters are given in Table 1.

H ¼ : = ð T :T Þ:

þ Q þ H ¼ R:

ð14Þ

Mvir is the virial mass, and crit (z) ¼ 3H(z)2 =8G

where the numerical constants are selected to correspond to 30% solar metallicity (Tozzi & Norman 2001; Ruszkowski & Begelman 2002). We assume spherical symmetry and solve the energy equation

ð12Þ

where rs is the standard scale radius of the NFW profile,   c3 3 200 ; ð13Þ M0 ¼ 2crit (z)r s 3 ln (1 þ c)  c=(1 þ c)

ð18Þ

ð17Þ

where ne is the electron density, T is the temperature, and ln C is the Coulomb logarithm. The suppression factor represents

where 0 ¼ e mH ne (0):

ð24Þ

Here e is the mean molecular weight per electron (we use e ¼ 1:19), and mH is the mass of hydrogen atom. The results for these fits are shown in Figure 1. Because the function vanishes at the origin, equation (20) is singular there. The boundary value for the velocity at r ¼ 0 is determined by assuming that the derivative is finite at the origin and setting the first term in brackets on the right-hand side of equation (20) equal to zero. The derivative of u at the origin is

Fig. 1.—Temperature and electron-density fits for the clusters A1795, A2597, and A478.

Fig. 2.—Plot of rms turbulent velocity profiles for the three clusters for three choices of the parameter  (top row). Corresponding normalized heating rates from diffusion (Q), dissipation ( ), and conduction (H ) for the same three choices of  (subsequent rows), where R is the rate of radiative cooling.

210

Fig. 3.—Velocity vs.  for two choices of radius and for cases in which the heating comes from only diffusion and conduction (Q, H ), only dissipation and conduction ( , H ), and from diffusion, dissipation, and conduction (Q, , H ). The straight line indicates the transition between weakly and strongly turbulent radial motions, which arises because of the effects of buoyancy.

211

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DENNIS & CHANDRAN

Fig. 4.—Top row: Velocity profiles with buoyancy (c0 6¼ 0) and without buoyancy (c0 ¼ 0). Bottom row: Velocity at fixed location as a function of , with and without buoyancy; short-dashed lines give u proportional to  1/3, and dash-dotted lines give u proportional to 1.

obtained by expanding each function as a Taylor series about the origin. 4. RESULTS In the top row of Figure 2, we present the velocity as a function of r for several choices of the parameter  in the range 0 <   1 for the three clusters. In subsequent rows, we show the fractional contributions of each heating source, once again including results for each value of  chosen. In each case, we obtain velocities in the range 100 P u P 300 kpc. The velocities tend toward larger values for larger values of  . For the two warmer clusters, A478 and A1795, dissipation exceeds turbulent diffusion when  P 0:5, and turbulent diffusion exceeds dissipation when  k 0:5. In the case of the cooler cluster A2597, dissipation exceeds turbulent diffusion for r k 80 kpc even for  ¼ 1:0, although turbulent diffusion exceeds dissipation in the central 50 kpc for  k 0:5. Interestingly, the heating from turbulent diffusion in A2597 changes sign and actually acts to cool the plasma at sufficiently large radii. We present in Figure 3 plots of velocity as a function of  for the three clusters at two different radii (r ¼ 20 and r ¼ 100 kpc). In each plot, we present results obtained for three different cases: (1) the model described in x 3, in which þ Q þ H ¼ R, (2) a revised model that neglects dissipation and in which Q þ H ¼ R, and (3) a revised model that neglects turbulent diffusion and in which þ H ¼ R. Also included in these plots is the line u ¼ c0 NBV ; l

ð25Þ

which marks the transition between the strongly and weakly turbulent parts of the u-l plane. For velocities near and below the line corresponding to equation (25), buoyancy leads to a significant reduction in turbulent diffusion, as described by equation (8). We see that for  P 0:5, neglecting dissipation of turbulence leads to a large increase in the value of u needed to balance cooling. On the other hand, for  k 0:5 the value of u increases by only a moderate amount if one of the two turbulent heating mechanisms is excluded. Additionally, we note that at r ¼ 20 kpc, for  k 0:4, the turbulence is at least moderately weak (u=l P NBV ), while at r ¼ 100 kpc the turbulence is at least moderately strong (u=l k NBV ) for all  in the range 0 <   1. To assess the role that buoyancy plays in the determination of the strength of the turbulence, we next examine how our results would differ if the NBV - dependent term in the expression for Deddy were neglected. In the top row of Figure 4, we plot velocity profiles for the three clusters for  ¼ 0:5 with and without the buoyancy term included. Since the effect of including buoyancy is to decrease the efficiency of turbulent diffusion through a reduction in the value of Deddy, we expect, and obtain, larger turbulent velocities to compensate for this loss of efficiency. In the second row of Figure 4 we again show how the velocity at a fixed distance (20 kpc) from the cluster center varies with ; this time with and without buoyancy and over a larger range of  . From equations (1), (8), and (9), we expect u  1 for large  when buoyancy is neglected (since in this limit Q /  and diffusion dominates dissipation) and u  1=3 for large  when buoyancy is included (since in this case Q and are both proportional to u3 =). This behavior is evident in Figure 4. For very small  , dissipation dominates over diffusion,

Fig. 5.—Plot of rms turbulent velocity profiles for the three clusters for three choices of the parameter f (top row). Corresponding normalized heating rates from diffusion (Q), dissipation ( ), and conduction (H ) for the same three choices of f (subsequent rows), where R is the rate of radiative cooling.

Fig. 6.—Ratio of dissipation to radiative cooling for several choices of  for r ¼ 20 kpc (left panels) and for r ¼ 100 kpc (right panels).

TURBULENT HEATING OF GALAXY-CLUSTER PLASMAS and it makes little difference whether or not buoyancy effects are included. In Figure 5, we present a series of plots similar to Figure 2, except that we fix  at the value 0.5 and vary f over the range 0:2 < f P fmax , where by fmax we refer to the largest possible value of f for which nonnegative velocities are obtained over the entire range of integration. (For A478 and A1795, fmax  0:4, while for A2597, fmax  1:3.) In the top row of Figure 5, we once again obtain velocity profiles in the range 100 < u < 300 km s1. As one would expect, higher velocities are associated with lower values of f. As f is increased to fmax, we find that the region within which conduction is the major source of heating becomes larger, but that regions remain at large and small radii where turbulent sources continue to dominate. As before, the cooler cluster A2597 exhibits qualitatively different behavior in that conduction makes its major contribution at large radii, and diffusion once again locally cools the plasma at large r. In Figure 6, we plot the ratio of dissipation to cooling =R as a function of velocity in all three clusters for several values of  at both r ¼ 20 and r ¼ 100 kpc. In every case, we see that as u varies from a small fraction of the sound speed, cs, to a value comparable to cs, the dissipation rate varies from a value far too weak to balance radiative cooling to a value far in excess of radiative cooling. This plot emphasizes the point that if turbulent heating and thermal conduction in fact balance radiative cooling in clusters, some explanation is needed for the finetuning of u to the required value.

215

A1795, and A2597. We solve for the rms turbulent velocity u as a function of radius r by setting þ Q þ H ¼ R, where , Q, and H are the heating rates from dissipation of turbulence, turbulent diffusion, and thermal conduction, respectively, and R is the rate of radiative cooling. We take the thermal conductivity to be a fixed fraction (typically one-fifth) of the Spitzer value for a nonmagnetized plasma (Spitzer 1962) and set the dominant velocity length scale l equal to r þ l0 , where  is a freeparameter constant, and l0 ¼ 0:5 kpc. We neglect nonthermal pressure and account for the effects of buoyancy forces on turbulent diffusion. For 0:05 <  < 1, we find velocities in the range 100 P u P 300 km s1. Roughly speaking, we find that k Q when  < 0:5, and P Q when  > 0:5, although there are exceptions to this rule. In general, both dissipation and diffusion are important. The inclusion of dissipation substantially reduces the value of u needed to balance cooling when  P 0:5, relative to models in which turbulent diffusion is the only form of turbulent heating. The inclusion of buoyancy effects leads to an increase in u, relative to models in which buoyancy is neglected. The magnitude of this increase is moderate for  ¼ 0:5 and large for  > 1. We also find that as u is varied from 0.1cs to cs, where cs is the sound speed, the ratio =R varies from a value much less than 1 to a value much greater than 1. This highlights the point that if turbulent heating approximately balances radiative cooling in clusters, an explanation is needed for the fine-tuning of u to the required value.

5. SUMMARY

The authors thank Eric Blackman, Axel Brandenburg, and Jason Maron for helpful discussions and Steve Allen, Brian McNamara, and Stefano Ettori for providing the data for A478, A1795, and A2597 and the anonymous referee for a very helpful report. This work was supported by NSF grant AST-0098086 and DOE grants DE-FG02-01ER54658 and DE-FC02-01ER54651 at the University of Iowa.

In this paper, we present a heated cooling-flow model in which radiative cooling is everywhere balanced by heating from viscous dissipation of turbulent motions, turbulent diffusion of heat, and thermal conduction. We take the density and temperature to be known functions of radius r obtained by fitting analytic functions to observations of three clusters: A478,

APPENDIX RELATION BETWEEN l AND lI The integral length scale is given by (Batchelor 1953) lI ¼ where u2 ¼ 2

R1 0

3 2u2

Z

1

dk 0

E(k) ; k

E(k) dk. If the kinetic energy spectrum is given by ( (k=kp )5=3 E(k) ¼ constant ; (k=kp )s

ðA1Þ

if k > kp ; if k < kp ;

ðA2Þ

then    3(s þ 1) lI ¼ : kp 10s

ðA3Þ

It is generally believed that s ¼ 2 is correct for the large-scale spectrum (Saffman 1967; Skrbek & Stalp 2000; Haugen et al. 2004), which yields lI ¼

9  : 20 kp

ðA4Þ

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