Cooling Turbulence Wiki Page

Back of the envelope

Isotropic forcing - or acceleration \mathcal{A} at a length scale \mathcal{L} of material at a density \rho will drive turbulent motions at a velocity \mathcal{V}=\sqrt{\mathcal{A}\mathcal{L}} that will dissipate in a crossing time t_{dyn}=\mathcal{L}/\mathcal{V} giving a heating rate \mathcal{H}=\rho\mathcal{V}^2/t_{dyn}=\rho\mathcal{V}^3/\mathcal{L}=\rho\mathcal{A}^{3/2}\mathcal{L}^{1/2}

In the absence of cooling, the turbulence would raise the temperature to the kinetic temperature of the turbulence in one crossing time - and the turbulence would become sub-sonic.

However, balancing this heating is a cooling term \mathcal{C}=\rho^2 \lambda(T).

In addition to the dynamical time, we now have a cooling time for the shocked material. t_{cool}=\frac{3k_BT_s}{\rho X \lambda(T_s)}.

Of key importance is the dimensionless ratio R=\frac{t_{cool}}{t_{dyn}}=\frac{\mathcal{V}^3}{\rho \mathcal{L} \lambda(\frac{X \mathcal{V}^2}{k_B})} = \frac{\lambda_{turb}}{\lambda(T_s)} where \lambda_{turb}=\frac{\mathcal{A}^{3/2}\mathcal{L}^{1/2}}{\rho}

If the cooling time is much less than the dynamical time - then shocked gas will quickly cool and the shocks will seem isothermal. If the cooling time is much longer than the dynamical time - then the gas as a whole will heat up.

If we plot this ratio as a function of turbulence velocity and the product of the (number) density and the driving scale R=R(\mathcal{V},(\rho \mathcal{L})) we get the following image

Contour Plot

Note we've labelled our three runs 'A','B', & 'C'

We can also consider the combined effect of the global heating by the turbulence and the global cooling of the ambient gas. Combining the heating and cooling we arrive at \frac{\rho}{X} k_B\dot{T}=\mathcal{H}-\mathcal{C} which simplifies to \dot{T}=\frac{\rho X}{k_B}\left(\lambda_{turb} - \lambda(T) \right). This gives an equilibrium temperature of T_{eq}=\lambda^{-1}(\lambda_{turb}) for 1e21.0 < \lambda_{turb| < 1e26.5 and a relaxation time of t_{relax}=\frac{k_B T}{\rho X \left( \lambda_{turb} - \lambda(T) \right)}. Assuming that \lambda(T) is an increasing function of T which it is for 1e2 < T < 1e5.3 the system will always evolve towards T_{eq} where \lambda(T_{eq}) = \lambda_{turb}.

Finally once the turbulent gas has reached the equilibrium temperature, the ratio \frac{T_s}{T_{eq}} will determine whether the turbulence will ultimately be supersonic or subsonic. We can characterize this by \mathcal{M}_{eq} = \frac{\mathcal{V}}{c_{eq}} \mbox{ where } c_{eq}=\sqrt{\frac{\gamma k_B T_{eq}}{X}}

The following image shows the same contours as before although over an extended range in the y-direction. Behind the contours however is a color map showing the mach number of the turbulence at the equilibrium temperature. The missing values imply that the turbulent \lambda is outside of the range of the cooling function - and no equilibrium solution exists…

Contours with Equilibrium Mach Number

Results

Results

Running three preliminary runs…

Description	Run	R=\frac{t_{cool}}{t_{dyn}}	\mathcal{M}_{eq}=\frac{\mathcal{V}}{c_{eq}}	n_{amb} cc$^{-1}$	T_{amb} (K)	\mathcal{L} (AU)	\mathcal{V} (km/s)	T_{eq} (K)	\lambda_{turb} (erg s$^{-1}$ cm$^{-6}$)	t_{cool} (yr)	t_{dyn} (yr)
Too Hot	A	2.8e3	.28	1.0e1	1.0e4	100	18.2	9.6e4	6.6e-22	7.3e5	2.6e2
Too Cold	B	1.0e-2	2.9	2.5e3	1.6e5	400	72.9	1.4e4	4.3e-23	6.6e2	6.4e4
Just Right	C	8.5e-1	.55	2.0e1	4.0e4	400	36.5	9.8e4	6.7e-22	8.8e2	1.0e3
High Mach	D	1.0e-5	10.	1.6e4	4.2e4	4e5	37.5	293	9.1e-27	8.3e2	8.0e7

Run A

The first run was well below the line where the cooling time equals the crossing time. By frame 35 the ambient has heated up and by frame 118 - it is clear that everything is hot and subsonic.

Frame 35

Frame 118

Run B

Run B is strongly cooling.

Frame 35

Run C

Run C is moderately cooling.

Frame 35

Column Density

(​movie)

Spectra

Frame 154

This image shows the velocity spectra (left) and density spectra (right) for runs A (red), B (green), and C (blue). (​movie)

PDFs

The image below shows 2D joint pdf's of [v vs. T] (top row), [rho vs. T] (middle row), & [rho vs. v] (bottom row) for Runs A (left column), B (middle column), & C (right column). (movie​)