DENSITY POWER SPECTRUM OF COMPRESSIBLE HYDRODYNAMIC TURBULENT FLOWS. Jongsoo Kim, KoreaAstronomy and Space Science Institute, Hwaam-Dong, Yuseong-Gu, Daejeon 305-348, Korea, ([email protected]), Dongsu Ryu,Department of Astronomy and Space Science, Chungman National University, Daejeon 305-764, Korea, ([email protected]).
According to the currently accepted paradigm, the inter-stellar medium (ISM) is in a state of turbulence and the turbu-lence is believed to play an important role in shaping complexstructures of velocity and density distributions [17, 6]. TheISM turbulence is transonic or supersonic with Mach numbervarying from place to place; while the turbulent Mach number,� , is probably of order unity in the warm ionized medium(WIM) judging from the temperature that ranges from ����
�
to ��� K [8], it is a few in the cold neutral medium (CNM)
[10] and as large as 10 or higher in molecular clouds [14]. Thefact that � �
�� implies high compressibility, and observa-
tions of the ISM reflect the density structures resulted from theturbulence. Interpreting these observations and hence under-standing the density structures require a good knowledge ofdensity statistics.
Density power spectrum, ��, is a statistics that can bedirectly extracted from observations. The best-known densitypower spectrum of the ISM is the one presented in [1, 2]. It is acomposite power spectrum of electron density collected fromobservations of velocity fluctuations of the interstellar gas, ro-tation measures, dispersion measures, interstellar scintillationsand others. What is remarkable is that a single Kolmogorovslope of ���� fits the power spectrum for the wavenumbersspanning more than 10 decades. But there are also a number ofobservations that indicate shallower spectra. For instance, thedensity power spectrum of cold H I gas has a much shallowerslope of����� � ���� [5]. Here we note that the power spec-trum in [1, 2] should reveal density fluctuations preferably inthe WIM since it is for the electron density. On the other hand,the power spectrum in [5] represents the so-called tiny-scaleatomic structure in the CNM [9].
Discussions on density power spectrum in the context oftheoretical works have been scarce. It is partly because thestudy of turbulence was initiated in the incompressible limit.Instead velocity power spectrum was discussed extensively bycomparing the well-known theoretical spectra, such as thoseof Kolmogorov [12] and Goldreich & Sridhar [7]. Hence, al-though hydrodynamic or magnetohydrodynamic (MHD) sim-ulations of compressible turbulence were performed, the spec-tral analysis was focused mainly on velocity (and magneticfield in MHDs) [4, 18]. A few recent simulation studies, how-ever, have reported density power spectrum [16, 13, 3]. Forinstance, in an effort to constrain the average magnetic fieldstrength in molecular cloud complexes, Padoan et al. [16]have simulated MHD turbulence of root-mean-square sonicMach number ���� � �� with weak and equipartition mag-netic fields. They have found that the slope of density powerspectrum is steeper in the weak, super-Alfvenic case than inthe equipartition case, and super-Alfvenic turbulence may bemore consistent with observations. Although their work hasfocused on the dependence of the slope of density power spec-trum on Alfvenic Mach number, they have also demonstrated
that the slope is much shallower than the Kolmogorov slope.Kritsuk et al. [13] and Beresnyak et al. [3] have found consis-tently shallow slopes. However, none of the above studies haveyet systematically investigated the dependency of the slope ofdensity power spectrum on sonic Mach number.
In this paper we study the density distribution and thedensity power spectrum in transonic and supersonic turbu-lent flows using three-dimensional hydrodynamic simulations.Specifically we show that the Mach number is an importantparameter that characterizes the density power spectrum.
Isothermal, compressible hydrodynamic turbulence wassimulated using a code based on the total variation diminishingscheme [11]. Starting from an initially uniform medium withdensity �� and isothermal sound speed � in a box of size �,the turbulence was driven, following the usual procedure [15];velocity perturbations were added at every �� � ��������,which were drawn from a Gaussian random field determined bythe top-hat power distribution in a narrow wavenumber rangeof � � � � �. The dimensionless wavenumber is defined as� � ��, which counts the number of waves with wavelength inside �. The amplitude of perturbations was fixed in sucha way that ���� of the resulting flows ranges approximatelyfrom 1 to 10. Self-gravity was ignored. Three-dimensionalnumerical simulations was performed with ���� grid zones.
Left and center panels in Figure 1 show the density dis-tribution in a two-dimensional slice for a transonic turbu-lence with ���� � ��� and a supersonic turbulence with���� � ��, respectively. The images reveal different mor-phologies for the two cases. The transonic image includescurves of discontinuities, which are surfaces of shocks withdensity jump of a few. So the three-dimensional density distri-bution should contain surfaces of discontinuities on the top ofsmooth turbulent background. On the other hand, the super-sonic image shows mostly density concentrations of string anddot shapes, which are sheets and filaments in three-dimension.Hence, the three-dimensional density distribution should bedominated by sheets and filaments of high density concentra-tion.
The right panel of Figure 1 shows the time-averaged den-sity power spectra for flows with different ����’s, along withfitted values of slope. The slope for the transonic turbulencewith ���� � ��� is�����, close to the Kolmogorov value of����. It is because the power spectrum represents the fluc-tuations of turbulent background, rather than discontinuitiesof weak shocks. Note that, in three dimension space, turbu-lence has the rotational mode of eddy motions in addition tothe compressional mode of sound waves and shock discon-tinuities, and the normal cascade is allowed. In supersonicturbulence, we get the slopes of ����, ����� and ����� for���� � ��, ��� and ����, respectively. The shallow spec-trum reflects the highly concentrated density distribution, butfor this time, of sheet and filament morphology shown in the
Protostars and Planets V 2005 8277.pdf
Figure 1: Three-dimensional turbulence. Left and center panels: Color images of the density distribution in a two-dimensionalslice, after turbulence was fully saturated, for a transonic turbulence with time averaged ���� � ��� (left) and a supersonicturbulence with time averaged ���� � �� (center). The color is coded on linear scales and the range of density is an order ofmagnitude larger for the supersonic case than for the transonic case. Right panel: Statistical error bars of the time-averaged densitypower spectra for flows with different ����’s. Solid lines and their slopes, which are obtained by least-square fits over the rangeof �
� � �� �, are included. Simulations used ���� grids zones.
center panel. Like delta-functions, infinitely thin sheets andfilaments in three-dimension give �� � ��.
In the first paragraph, we have pointed that in the ISMthe power spectrum of electron density has the Kolmogorovslope [1, 2], whereas the power spectrum of H I gas showsa shallower slope [5]. Our results suggest the reconciliationof these claims of different spectral slopes. That is, the elec-tron density power spectrum represents the density fluctuationsmostly in the WIM, where the turbulence is transonic and thedensity power spectrum has the Kolmogorov slope. On theother hand, the H I power spectrum represents the tiny-scaleatomic structures in the CNM, where the turbulence is super-sonic with ���� � a few and the density power spectrum hasa shallower slope.
References
[1] Armstrong, J. W., Cordes, J. M. & Rickett, B. J. 1981,Nature, 291, 561
[2] Armstrong, J. W., Rickett, B. J. & Spangler, S. R. 1995,ApJ, 443, 209
[3] Beresnyak, A., Lazarian, A. & Cho, J. 2005, ApJ, 624,L93
[4] Cho, J. & Lazarian, A. 2002, Phys. Rev. Lett., 88, 245001
[5] Deshpande, A. A., Dwarakanath, K. S. & Goss, W. M.2000, ApJ, 543, 227
[6] Elmegreen, B. G. & Scalo, J. 2004, ARA&A, 42, 211
[7] Goldreich, P. & Sridhar, H. ApJ, 438, 763[8] Haffner, L. M., Reynolds, R. J. & Tufte, S. L. 1999, ApJ,
523, 223
[9] Heiles, C. 1997, ApJ, 481, 193
[10] Heiles, C. & Troland, T. H. 2003, ApJ, 586, 1067
[11] Kim, J., Ryu, D., Jones, T. W. & Hong, S. S. 1999, ApJ,514, 506
[12] Kolmogorov, A. 1941, Dokl. Akad. Nauk SSSR, 31, 538
[13] Kritsuk, A., Norman, M. & Padoan, P. 2004 (astro-ph0411626)
[14] Larson, R. B. 1981, MNRAS, 194, 809
[15] Mac Low, M.-M. 1999, ApJ, 524, 169
[16] Padoan, P., Jimenez, R., Juvela, M. & Nordlund, A. 2004,ApJ, 604, L49
[17] Vazquez-Semadeni, E., Ostriker, E. C., Passot, T., Gam-mie, C. F. & Stone, J. M. 2000, in Protostars and PlanetsIV, eds. V, Mannings, A. P. Boss & S. S. Russell (Tuscon:University of Arizona Press), p 3
[18] Vestuto, J. G., Ostriker, E. C. & Stone, J. M. 2003, ApJ,590, 858
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