§34 Discontinuities in jet-cloud collisions

As it was discussed in Chapter V the one dimensional collision between a shock and a cloud generates two types of strong discontinuities: shock waves and rarefaction waves. It still remains an open question as to how good these solutions are for understanding the more general three dimensional case. At least it should be expected that far away from the boundaries of a spherical cloud, where the interaction can be considered plane-parallel the solution should not differ too much from the one dimensional case.

When the two dimensional case is considered, more strong discontinuities (shocks) are generated and the cloud is unstable due to the generation of Raleigh-Taylor and Kelvin-Helmholtz instabilities (Klein et al., 1994). This is seen analytically as follows.

When a shock wave propagates through the interstellar or intergalactic medium with a velocity $\ensuremath{v}_b$ and collides with a cloud, it drives a shock wave into it (cf. Chapter V). Let us assume that the incoming shock is strong so that the Mach number $M \gg 1$. Under this circumstances, the postshock pressure is about $\rho_e \ensuremath{v}_b^2$ (Landau & Lifshitz, 1995), with $\rho_0$ being the density of the external medium around the cloud. In the same way, the pressure behind the shock in the cloud is about $\rho_c \ensuremath{v}_s^2$, where $\rho_c$    is the original density of the cloud, and $\ensuremath{v}_s$ is the the shock velocity inside the cloud. These two pressures most be comparable, and so (Klein et al., 1994; McKee & Cowie, 1975; Bychkov & Pikelner, 1975):

$$\ensuremath{v}_s \simeq \frac{ \ensuremath{v}_b }{ \sqrt*{ \chi } },$$ (34.1)

where $\chi \equiv \rho_c / \rho_e$ is the density contrast between the cloud and the intercloud environment. The time for the shock in the intercloud medium to sweep across the cloud is:

$$t_{\textrm{ic}} \equiv \frac{ 2 \Delta_0 }{ \ensuremath{v}_b },$$ (34.2)

in which $\Delta_0$ is the initial radius of the cloud. The characteristic time for the cloud to be ``crushed'' by the shocks inside the cloud is $\Delta_0 / \ensuremath{v}_s$. From this and eq.(34.1) it follows that the cloud crushing time is:

$$t_{ \textrm{cc} } \equiv \frac{ \chi^{1/2} \Delta_0 }{ \ensuremath{v}_b }.$$ (34.3)

After the blast wave has swept over the cloud, the shocked cloud is subject to both, Kelvin-Helmholtz and Rayleigh-Taylor instabilities (Klein et al., 1994). For $\chi \gg 1$ the timescale $t_{\textrm{KH}}$ for the growth of Kelvin-Helmholtz instabilities for perturbations of wavenumber $k$ parallel to the relative velocity $\ensuremath{v}_{\textrm{rel}}$ between the cloud and intercloud media is $t^{-1}_{\textrm{KH}} = k \ensuremath{v}_{\textrm{rel}} / \chi^{1/2}$ (Klein et al., 1994; Chandrasekhar, 1961). In other words, the Kelvin-Helmholtz growth time is comparable to the cloud crushing time:

$$\frac{ t_{\textrm{KH}} }{ t_{\textrm{cc}} } = \frac{ \ensuremath{v}_b / \ensuremath{v}_{\textrm{rel}} }{ k \Delta_0 }.$$ (34.4)

Short wavelengths have a fast growth, but longer wavelengths ( $k \Delta_0 \sim 1$) are far more disruptive.

The blast wave accelerates the cloud in two stages (Klein et al., 1994). Firstly, the cloud shock accelerates it to a velocity $\ensuremath{v}_s$. Secondly, the flow of shocked intracloud gas accelerates it until it is comoving with the flow behind the incoming shock, which for a monoatomic gas has a velocity of $3 \ensuremath{v}_b / 4$. For a large density contrast, the cloud shock velocity is small and the acceleration is dominated by the second stage. Let $\ensuremath{v}_c$ be the mean velocity of the cloud, $\ensuremath{v}_{\textrm{i1}}$ the velocity of the shocked intercloud medium and $\ensuremath{v}'_c = \vert \ensuremath{v}_{\textrm{i1}} - \ensuremath{v}_c \vert$ the magnitude of the velocity of the cloud relative to the shocked intercloud medium. With these, the equation of motion of the cloud can be written (Klein et al., 1994; Landau & Lifshitz, 1995):

$$m_c \frac{ \mathrm{d} \ensuremath{v}'_c }{ \mathrm{d} t } = - \frac{ 1 }{ 2 } C_D \rho_{\textrm{i1}} {\ensuremath{v}'_c}^2 A,$$ (34.5)

where $m_c$ is the mass of the cloud, $C_D \sim 1$ is the drag coefficient, $\rho_{\textrm{i1}} \sim 4 \rho_0$ is the density of the shocked intercloud medium and $A$ is the cross section area of the cloud. If this cross section remains constant then $A \approx \pi \Delta_0^2$ and the mass of the cloud $m_c \approx \rho_c V_c$ in which the volume of the cloud $V_c \propto \pi \Delta_0^3$. Since the velocity $\ensuremath{v}'_c \sim \ensuremath{v}_b$, it follows that eq.(34.5) defines a characteristic drag time $t_{\textrm{drag,0}}$ for a strong shock given by (Klein et al., 1994):

$$t_{\textrm{drag,0}} = \frac{ \chi \Delta_0 }{ C_D \ensuremath{v}_b } = \frac{ \chi^{1/2} t_{\text{cc}} }{ C_D }.$$ (34.6)

The deceleration of the cloud initially proceeds on the drag timescale $t_{\textrm{drag,0}}$. This gives a deceleration $g \simeq \ensuremath{v}_b / t_{\textrm{drag,0}}$, which corresponds to a Rayleigh-Taylor growth time given by $t^{-1}_{\textrm{RT}} \simeq \sqrt{ gk }$ (Chandrasekhar, 1961). This instability has a growth time of the order of the cloud crushing time (Klein et al., 1994):

$$\frac{ t_{\textrm{RT}} }{ t_{\textrm{cc} } } \simeq \frac{ 1 }{ \left( k \Delta_0 \right)^{1/2} }.$$ (34.7)

The above results suggest that the cloud will be destroyed in a time related to the cloud crushing time. In fact, numerical simulations carried out by Nittmann (1982) and Klein (1994) show that the cloud destruction time is indeed a few times the cloud crushing time.

The two dimensional problem of the collision between a shock wave and a cylindrical cloud of constant density can be described as follows according to numerical simulations (Klein et al., 1994). Four stages can be identified in the collision.

1. When the incoming plane parallel shock wave hits the cylindrical cloud two shocks are formed. One penetrates the cloud and another one is reflected from the point of impact (cf. Chapter V). The reflected shock settles into a standing bow shock in a time of the order $\Delta_0 / \ensuremath{v}_b = t_{\textrm{ic}} / 2$.
2. The next stage is shock compression of the cloud. After a time $\sim t_{\textrm{ic}}$ the flow around the cloud converges on the axis behind the cloud, producing a high pressure reflected shock in the intercloud medium and driving a shock into the rear of the cloud. The shocks produced at the sides of the cloud are weaker than those at the front and the back of the cloud. This is due to the fact that the pressure is a minimum at the sides of the cloud. As a result of all this, the cloud is compressed into a thin pancake, with its traverse dimension reduced by a factor of 2. The collision of the transmitted shock into the cloud with the shock coming from the rear produces further compression.
3. When the shock transmitted into the cloud reaches the opposite side of the cloud it transmits a shock to the intracloud medium and reflects back a rarefaction wave. This produces the so called reexpansion stage. At the same time, the low pressure at the sides of the cloud compared to that on the axis causes the cloud to expand laterally. This lateral expansion continues up to a time of a few cloud crushing times.
4. Finally, instabilities and differential forces due to the flow of the intercloud gas past the cloud makes the last stage: cloud destruction. This causes the cloud to fragment.

As it is seen from the above discussion and from the results presented in Chapter V, the one dimensional case and the two dimensional one are quite different. This is mainly due to the fact that in the two dimensional case the cloud is of a finite size (cylindrical cloud) whereas in the one dimensional case it is not (plane parallel cloud). However, the most important strong discontinuities (shocks and rarefaction waves) are produced in both cases. From the results of Chapter V it follows that the pressure behind the transmitted shock (for a large density contrast) should be about six times the pressure in the intercloud medium, as would be expected from the reflection of a strong shock with a solid wall (Spitzer, 1982; McKee, 1988). However, such a high pressure has not been seen in the numerical simulations by (1990) and Klein (1994). It seems that this high pressure exists only as the flow is one dimensional or it does happen for such a short time that it does not show up in the simulation.

Despite all of these differences, the perturbed solution found in Chapter V serves as an upper limit to values of pressures, densities and temperatures and provides a very simple way of understanding the behaviour of the collision between a shock wave and a cloud in detail.

An important use of the results presented in Chapter V is to compare the cooling time behind the transmitted shock wave to the shock crossing time. Let us assume that the density contrast is $10^4$ (see section §6) and that the shock that penetrates the cloud after the first initial discontinuity compresses the gas $\sim 5 \times 10^2$ of its original value, as shown pictorically in fig.(V.5). The density in the cloud increases by a factor of $\sim 3$ with the passage of the shock wave. The temperature of the cloud is enhanced by a factor of $\sim 1.6 \times 10^4$ of its original value. In other words, if we assume that the temperature of the cold clouds is $\sim \unit{ \power{ 10 }{ 4 } }{ \kelvin }$, then the passage of the shock wave is able to increase this value to $\sim \unit{ \power{ 10 }{ 8 } }{ \kelvin }$. The dissipation of the heated gas in the cloud, that is, its energy loss produced by radiation can be calculated from calculations made by (1993); (1993) and described by (1998). If the cooling function $\Lambda$ is measured in $\unit{}{ ergs \usk \centi \meter \cubed \usk \reciprocal \second }$, then $\mathrm{d} E / \mathrm{d} t = -n^2 \Lambda(T)$ is the energy loss rate per unit volume in which $E$ is the energy of the system, $t$ time, $n$ the particle number and $T$ the temperature of the gas. From the results presented in (1998), it follows that $\Lambda(T \sim \! \unit{ \power{ 10 }{ 8 } }{ \kelvin }) \approx \unit{ \power{ 10 }{ -23 } }{ ergs \usk \centi \meter \cubed \usk \reciprocal \second }$. The cooling time $\tau_{\textrm{cool}} \equiv E / \vert \mathrm{d} E / \mathrm{d} t \vert$ is then given by $\tau_{\textrm{cool}} \sim \unit{ \power{ 10 }{ 3 } }{ yr }$, where we have used the fact that $E = 3 n \mathsf{k}_B T$ for a fully ionised gas, $\mathsf{k}_B$ is the Boltzmann constant and the particle number density of clouds needed for the radio alignment effect is $n_{\textrm{cloud}} \sim \unit{ \power{ 10 }{ 3 } }{ \centi \meter \rpcubed }$ (Best et al., 2000).

The time it takes the incoming shock with velocity $\ensuremath{v}_s$ to cross a cloud of width $\Delta$ is $\tau_{\textrm{cross}} = \Delta / \ensuremath{v}_s$. Typical values for the advance speed of a jet in a radio source are $\ensuremath{v}_s \sim 0.1 \ensuremath{\mathsf{c}}$. The characteristic size of a cloud needed to produce the observed radio-optical alignment effect is about $\unit{ 1 }{ \kilo pc }$. This implies that the crossing time is $\tau_{\textrm{cross}} \sim \unit{ \power{ 10 }{ 4 } }{ yr }$. In other words $\tau_{\textrm{cool}} \sim \tau_{\textrm{cross}}$, so thermal instabilities are likely to develop inside the cloud. Because of this, if a shock wave associated with an expanding jet in a radio source interacts with a set of clouds in the interstellar and intergalactic medium, it is possible that the optical emission observed in the alignment effect in radio galaxies is produced.