ESTIMATION OF MAGNETIC FIELD STRENGTH IN THE TURBULENT WARM IONIZED MEDIUM

The warm ionized medium (WIM) is one of the major gas components of our Galaxy. It is a diffuse ionized gas with temperature T ∼ 8000 K, scale height H ∼ 1 kpc, and average density ${\bar{n}} \sim 0.03$ cm⁻³, and occupies approximately 20% of the volume of the disk in the Galaxy (e.g., Reynolds 1991; Haffner et al. 1999). The measured width of the Hα line from the WIM is in the range of 15–50 km s⁻¹ (Tufte et al. 1999). Observations indicate that the WIM is in a turbulent state (see below). Assuming that the typical width of Hα line and temperature are 30 km s⁻¹ and 8000 K, respectively, the non-thermal, turbulent velocity would be about 13 km s⁻¹, as is calculated, for instance, with Equation (1) of Reynolds (1985). So the sonic Mach number of turbulent motions, M_s, would be around unity in the WIM. It is smaller than M_s of the cold neutral medium, which is a few (e.g., Heiles & Troland 2003), and M_s of molecular clouds, which is ≳10 (e.g., Larson 1981).

The best evidence for turbulence in the interstellar medium (ISM), which includes the WIM, comes from the power spectrum presented in Armstrong et al. (1995). It is a composite power spectrum of electron density collected from observations of velocity fluctuations of the interstellar gas, rotation measures (RMs), dispersion measures (DMs), interstellar scintillations, and others. The spectrum covers the spatial range of ∼10¹⁰–10²⁰ cm, and remarkably the whole spectrum is fitted to the power spectrum of Kolmogorov turbulence with slope −5/3. In addition, it has been recently reported that the Hα emission measure for the WIM (Hill et al. 2008) and the densities of the diffuse ionized gas and the diffuse atomic gas (Berkhuijsen & Fletcher 2008) follow the lognormal distribution. The lognormality in density (or column density) distributions can be regarded as another signature of turbulence in the WIM (e.g., Vázquez-Semadeni 1994; Elmegreen & Scalo 2004, and references therein).

The interstellar magnetic field that is pervasive in the Galaxy plays an important role in the dynamics of the ISM, star formation, acceleration of cosmic rays, etc. It has the energy density comparable to those of turbulence and cosmic rays, as well as the thermal energy density (Spitzer 1978). The information on the field has been obtained through observations of Zeeman splitting, polarized thermal emission from dust, optical starlight polarization, radio synchrotron emission, and the Faraday rotation of polarized radio sources (see Han & Wielebinski 2002 for a review). Of them, the last method can be used to study the magnetic field in ionized media such as the WIM. It estimates the mean strength of the magnetic field along the line of sight, weighted with the electron density n_e,

$$\begin{equation} \langle B_{\parallel }\rangle = \frac{\int n_{e} B_{\parallel } ds}{\int n_e ds} \equiv \frac{\rm RM}{\rm DM}, \end{equation} \tag{ 1 }$$

where B_∥ ≡ Bcos θ and θ is the angle between B and the line of sight. Several authors (e.g., Han et al. 1998; Indrani & Deshpande 1999; Frick et al. 2001; Han et al. 2006; Beck 2007) have used the RMs and DMs of pulsars to reproduce the large-scale magnetic field of our Galaxy.

Recently, the distributions of RMs along many contiguous lines of sight have been obtained in multi-frequency polarimetric observations of the diffuse Galactic synchrotron background (Haverkorn et al. 2003, 2004; Schnitzeler et al. 2009) and the Perseus cluster (de Bruyn & Brentjens 2005). While the peak in the frequency distribution of the RMs reflects the regular component of the magnetic field, the spread should reflect the turbulent component. This means that if the distribution of RMs is observed, the spread provides another way to quantify the magnetic field in turbulent ionized media such as the WIM.

Motivated by the importance of RM in the study of the magnetic field in the WIM, in this Letter we study RM in turbulent media with the rms Mach number of unity. Simulations are outlined in Section 2. Results are presented in Sections 3 and 4.

We performed three-dimensional simulations of isothermal, compressible, magnetohydrodynamic (MHD) turbulence, using a code based on the total variation diminishing scheme (Kim et al. 1999). The fact that the temperature of the WIM is in a relatively narrow range of 6000–10000 K (Haffner et al. 1999) would justify the assumption of isothermality. There are two parameters in our simulations: the root-mean-square (rms) sonic Mach number, M_s, and the initial plasma beta value, β₀. Hereafter, the subscript "0" is used to denote the initial values. The first parameter indicates the level of turbulence. In this Letter, we focus on the turbulence with the rms Mach number of unity, and so set M_s = 1. The second parameter tells the strength of the initially uniform magnetic field, or the regular field, B₀. In order to explore the effect of the regular field, we varied the value of β₀ from 0.1, 1, 10, to 100. If we take 8000 K and 0.03 cm⁻³ as the representative values of the temperature and electron density, assuming that hydrogen is completely ionized, helium is neutral, and the number ratio of hydrogen to helium is 0.1, we have B₀ = 1.3(1/β₀)^1/2(T/8000 K)^1/2(n_e/0.03 cm⁻³)^1/2μG. So the initial magnetic field strength in our simulations corresponds to 4.1, 1.3, 0.41, 0.13 μG for β₀ = 0.1, 1, 10, and 100, respectively.

Simulations were started with B₀ along the x-direction in a uniform medium of density, ρ₀. The grid of 512³ zones was used for the periodic computational box of size L. Turbulence was driven with the recipe in Stone et al. (1999) and Mac Low (1999); velocity perturbations were drawn from a Gaussian random field determined by the top-hat power distribution in a narrow wavenumber range of (2π/L) ⩽ k ⩽ 2(2π/L), and added at every Δt = 0.001L/c_s. The amplitude of the perturbations was tuned in such a way that M_s ≡ v_rms/c_s is around unity at saturation. Here, c_s is the isothermal sound speed and v_rms is the rms velocity of the resultant turbulent flow. In simulations v_rms initially increased and became saturated at t ≃ (1/2)L/c_s, and we ran the simulations up to t = 2L/c_s, two sound crossing times. At saturation, the average strength of magnetic field reached 4.2, 1.45, 0.73, and 0.51 μG for β₀ = 0.1, 1, 10, and 100, respectively. The amplification factor was larger for the simulations with larger β₀ or weaker B₀.

We first check whether 〈B_∥〉 in Equation (1) correctly reproduces the unbiased strength of the magnetic field along the line of sight. That is true only if there is no correlation between B and n_e (or the gas density ρ). If the correlation is positive, 〈B_∥〉 in Equation (1) overestimates the magnetic field strength, while if negative, it underestimates the strength.

It is known from observations (Crutcher 1999; Padoan & Nordlund 1999) and numerical simulations (e.g., Ostriker et al. 2001; Passot & Vázquez-Semadeni 2003; Balsara & Kim 2005; Mac Low et al. 2005; Burkhart et al. 2009) that in the highly compressible, supersonic, molecular cloud environment, the magnetic pressure (or B) and the gas pressure (or ρ in isothermal gas) are positively correlated. Burkhart et al. (2009), on the other hand, reported a weak negative correlation for M_s = 0.7 and β₀ = 2 (their model 1). Figure 1 shows the correlation between B and ρ in our simulations. The nearly circular shape of contour lines indicates weak correlation for M_s = 1. To quantify it, we calculated the correlation coefficient

$$\begin{equation} r(B,\rho)= \frac{{\Sigma }_{i,j,k} (B_{i,j,k}-\bar{B})(\rho _{i,j,k}-\bar{\rho })}{[\Sigma _{i,j,k}(B_{i,j,k}-\bar{B})^2]^{1/2} [\Sigma _{i,j,k}(\rho _{i,j,k}-\bar{\rho })^2]^{1/2}}, \end{equation} \tag{ 2 }$$

where $\bar{B}$ and $\bar{\rho }$ are the average values of B and ρ. The values of r at the end of simulations were 0.01, −0.12, −0.06, and 0.16 for β₀ = 0.1, 1, 10, and 100, respectively. Relatively small values of r confirm that the correlation is weak for M_s = 1, regardless of β₀.

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Weak correlation means that the RM field in Equation (1) should correctly represent the true magnetic field. To test it, we compare the strengths of two fields in Figure 2; the frequency distributions of the true mean strength of the magnetic field along the line of sight (presented with solid lines) and the magnetic field strength calculated with Equation (1) (presented with dotted lines) coincide quite well. So we argue that the systematic bias, due to a correlation between B and n_e, in the estimation of magnetic field strength with RM would be insignificant in the turbulent media with M_s = 1.

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Beck et al. (2003) pointed that the discrepancy in the strengths of the regular Galactic magnetic field estimated with RM and synchrotron emissivity could be reconciled, if the correlation between B and n_e is negative in the WIM and so the RM field was underestimated. Beck et al. (2003) postulated the pressure equilibrium, which results in the negative correlation between B and ρ. If the pressure equilibrium is maintained, the combined pressure of gas and magnetic field, P_tot = P_gas + P_mag, should exhibit a narrow distribution. Figure 3 shows the frequency distribution of P_tot. Our simulations show broad distributions rather than peaked distributions. That is, our results do not support the prediction of Beck et al. (2003). However, we caution that we considered only the turbulent media with the rms Mach number M_s = 1. On the other hand, the WIM may have M_s which is not exactly unity. So we need to further investigate turbulent media with M_s ≠ 1, before we exclude the postulation of Beck et al. (2003). We leave it as a future work.

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We then look at the frequency distribution of RMs. Figure 4 shows the probability distribution of RM/$\overline{\rm RM}$ for different β₀'s as well as for different θ's. Here, $\overline{\rm RM}$ is the average value of RMs. (The distribution was also calculated for θ = 83°, but not shown in the figure for clarity. The distribution for θ = 83° is used in Figure 5.) The fit to the Gaussian is also shown. A noticeable point in the figure is that the distribution of RM/$\overline{\rm RM}$ is very well fitted to the Gaussian. The goodness-of-fit is between 0.89 and 0.99.⁵ This result is consistent with the observations of Haverkorn et al. (2003, 2004). They took the multi-frequency polarimetric images of diffuse radio synchrotron backgrounds in the constellation Auriga and Horologium, and obtained RM maps. The distribution of the observed RMs is also fitted to the Gaussian.

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Another noticeable point in Figure 4 is that the distribution is more widely spread for larger β₀ and larger θ. It hints at a possible correlation between the spread in the distribution of RM/$\overline{\rm RM}$ with the strength of the magnetic field along the line of sight. Figure 5 shows the relation between the full width at half maximum, W_FWHM, of the distribution of RM/$\overline{\rm RM}$ shown in Figure 4 and the strength of the regular field along the line of sight, B_0∥ ≡  B₀cos θ. The best fit for the relation⁶ is

$$\begin{equation} B_{0\parallel } = (2.45\pm 0.3) \times W_{\rm FWHM}^{-1.41\pm 0.1} \mu {\rm G}. \end{equation} \tag{ 3 }$$

Note that the relation is for T = 8000 K and n_e = 0.03 cm⁻³ as the representative values of the temperature and electron density, and scales as (T/8000 K)^1/2(n_e/0.03 cm⁻³)^1/2 for other values of T and n_e. The empirical relation in (3) may provide a handy way to quantify the strength of the regular field along the line of sight in regions where the map of RMs has been obtained along many contiguous lines of sight with the multi-frequency polarimetric observations (see the Introduction). The accompanying map of DMs is not necessary for it.

The estimation of magnetic field strength with the relation in (3), however, should be done with caution because of its limitations: first, the relation is valid only for M_s = 1. Second, no stratification of gas and magnetic field was assumed, which is apparent in the polarization maps of a larger area of our Galaxy. Third, no source emitting polarized light in the medium between background polarized continua and us was considered.

From a physical point of view, the broadening of the width of RM distribution is due to fluctuating magnetized gas. We can easily expect that, for a given regular field, the width would be larger if the rms Mach number, M_s, is larger. So W_FWHM should be a function of not only B_0∥ but also M_s. In order to quantify the relation of W_FWHM versus B_0∥ and M_s, we need simulations that cover the parameter space of β₀ and M_s. Low-resolution simulations with M_s = 0.5 and 2.0 show that the probability distribution of RM/$\overline{\rm RM}$ is still fitted to the Gaussian, and W_FWHM can differ by a factor of two to three if M_s differs by a factor of two. We leave the report of the M_s ≠ 1 results, including the correlation between B and ρ, from high-resolution simulations as a future work.

Nevertheless, we can try to apply the relation in (3) to RM observations in the WIM, assuming M_s = 1 there. For instance, from the figures of the frequency distribution of RMs in Auriga and Horologium in Haverkorn et al. (2003, 2004), we read W_FWHM ≃ 2 and 7, and get B_0∥ ∼ 0.6 and 0.1 μG for Auriga and Horologium, respectively (if n_e = 0.016 cm⁻³ is used as in Haverkorn et al. 2004). These are close to, but somewhat larger than, the values estimated in Haverkorn et al. (2004), ∼0.42 and 0.085 μG, respectively. However, by considering the limitations of the relation in (3) and the uncertainty in the values of W_FWHM which we read from figures, we regard the agreement to be fair.

Currently the number of the synchrotron backgrounds, which can be used for the observation of RM maps, is still limited due to the relatively low sensitivity of present-day radio telescopes. However, the new-generation radio telescopes, such as LOFAR (Low Frequency Array) and SKA (Square Kilometer Array) with much higher sensitivity, will certainly provide RM maps in much larger portions of the sky. Then, the relation like the one in (3) will become a useful diagnosis.

We thank R. Beck and anonymous referees for clarifying comments. The works of J.K., D.R., and J.C. were supported by the Korea Foundation for International Cooperation of Science and Technology through K20702020016-07E0200-01610. The work of J.K. was also supported by the National Research Foundation of Korea through 2009-0062863 (ARCSEC). The work of D.R. was also supported by the Korea Science and Engineering Foundation (KOSEF) through R01-2007-000-20196-0. This work utilized a high performance cluster built with fundings from the Korea Astronomy and Space Science Institute and KOSEF through the Astrophysical Research Center for the Structure and Evolution of Cosmos (ARCSEC).