Abstract
Models are the roadmaps through the observational thicket. Despite its enormous range properties, a relatively few basic physical processes govern much of the structure and evolution of the ISM. In this chapter, we review some of the most important. We begin with a reprise of the issue of virialization. Then we examine some instabilities to which the ISM is subject. The core of this chapter is the structural, energetic, and dynamical importance of large scale motions and turbulence in the diffuse medium and some of the issues we see as fruitful lines for further investigations.
I am an old man now, and when I die and go to Heaven there are two matters on which I hope enlightenment. One is quantum electrodynamics and the other is turbulence of fluids. About the former, I am really rather optimistic. Horace Lamb (author of Hydrodynamics and a referee of Reynold’s papers; cited by Goldstein, S. 1969, ARFM, 1, 23)
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Notes
- 1.
The first application of the full VT to the ISM, including magnetic fields and rotation, appears to have been Chandrasekhar and Fermi (1953) but it was already standard for treating stellar structure for decades before that.
- 2.
A simple way to see what happens if we include magnetism is to again think of the virial condition and set B 2 V ∼ GM 2∕R, where B is a mean value of the field regardless of its detailed configuration and the proportion allows for internal (structural) differences among arrangements of magnetic field and density. Then for the virial condition to hold (equilibrium with only the magnetic field opposing self-gravitation), so a very low β configuration [in a plasma, the ratio of the plasma pressure (P = nkT) to the magnetic pressure (P mag = B 2∕2μ o )], Φ m ∕M ≈ constant. where Φ m is the magnetic flux through a surface of radius R. Although such considerations are important for star forming clouds, this magnetized state is very far from what is observed in the diffuse medium.
- 3.
To see this last point, note that the derivative in the Eulerian version of the fluid equations contributes ik m δ mj to the interactions. Since the order of k and k′ is immaterial, and can be reversed, this is the same as saying that k+k′ = k″ and ω +ω′ = ω″ for the frequencies. This three wave coupling is when two colliding waves combine to produce a third at a combination frequency in space and time (Whitham 1974; Craink 1988). In turbulence, this condition is the basis of the nonlinear couplings (Sagdeev and Galeev 1969; McComb 1990; Pope 2002). But these conservation conditions may be inconsistent. The frequencies are scalar, the wavenumbers are vectors, and the frequency of the wave may be linked (for the weak turbulence case) directly to the wavenumbers. For instance, for Alfven waves, \(\omega = v_{A}\mathbf{k \cdot \hat{ B}_{0}}\) where the Alfven speed depends on the strength of the external field B 0. If two of these waves, moving at the same velocity, are oriented in the same direction because of a constant field strength they never interact if from different times. On the other hand, if oppositely directed along the field, they collide. What happens depends on the same condition as before. The frequencies add, the wavenumbers cancel, and if k was not initially present it cannot appear. The implication of this, realized in the earliest work on MHD turbulence by Iroshnikov (1963), Kraichnan (1966), and supported by Shebalin et al. (1983), Montgomery and Matthaeus (1995), and Ng and Bhattacharjee (1996) is that no parallel cascade is possible. The condition required for the K41 spectrum (Kolmogorov 1941a,) cannot occur. The waves interact through the distortion they produce in the field lines, something not available for normal fluid turbulence, and this coupling introduces an intrinsic anisotropy in the cascade. This is, nonetheless, only on a sort of microscale. It happens within the distance of a single wavelength for the scattering waves. Note that this is much smaller than the injection scale if we are within the range of the cascade. So likely it will not be seen, directly, in any imaging or dynamical probes. Its effect is, instead, to change the predicted spectrum and introduce a new scale for the lower end of the turbulence. The difference between weak and strong forms of turbulence depends, then, on the number of fluctuations it takes to change the velocity amplitude. The theories to date have been for incompressible turbulence. This works well for Alvfenic modes since they are strictly transverse (except when magnetosonic, fast, modes are included). The change in the amplitude is either a random walk of small increments, so varies like (t A ∕t dyn )2, or strong so it varies linearly with the ratio. The difference in the isotropy leads to a length scale, l ⊥ , that becomes a free parameter of the theory. To limit it, Goldreich and Sridhar (1995) introduced the concept of balance, that the rate of transfer between collisions on the parallel scale l is balanced by the rate of transfer among modes on the orthogonal scale.
- 4.
It is important to note that a confusion often arises in discussions of turbulent motions. The cascade derives its energy from driving on a scale that is much larger than the dissipation range. Numerical simulations actually have difficulty achieving this, especially shear flows, so instead they are often forced with gaussian stirring (but this is in time, not space). The spatial correlations are a distinctly non-gaussian feature. The randomness comes from events being independent in time but, wherever the fluctuation occurs, the coupling to different lengths establishes the cascade spectrum.
- 5.
The ensuing set of ideas are known as IK theory.
- 6.
On a technical point, this is because the picture presupposed the Burgers’ equation, the one dimensional Euler equation with a viscous dissipation term (Burgers, J.M. 1948, “A mathematical model illustrating the theory of turbulence”, Adv. Appl. Mech., 1, 171).
- 7.
In many numerical simulations of non-decaying turbulence, the driving is effected in the model by gaussian stirring based on the power spectrum of the cascade.
- 8.
A side comment is in order here. If the flow shows significant variations in multiple line tracers in two dimensions, the flow is certainly non-isotropic on any larger scale than that over which such differences are measurable. Even poorly sampled mapping will show this. Nonetheless, density and kinetic variations conspire to produce the line of sight optical depth that is what is really measured in channel maps. Were it possible to change our viewing angle through a structure, this ambiguity could be removed. Many of the statistical methods developed for galactic structure and stellar dynamics are like those we have described in this chapter. But stars are not a continuous medium. Proper motion of a point source is an actual, not apparent, displacement in the plane of the sky. The motion can be assigned to a single mass. This is not true for clouds or diffuse gas.
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Magnani, L., Shore, S.N. (2017). Dynamical Considerations: Instabilities and Turbulence. In: A Dirty Window. Astrophysics and Space Science Library, vol 442. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54350-4_11
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