# What does it mean for cold clouds to be in pressure equilibrium with a diffuse hot medium?

I often read about cold objects (namely cold clouds in the Galactic halo or cold filaments accreting into high-redshift galaxies) being in pressure equilibrium with the diffuse hot ambient gas. What does this pressure equilibrium intuitively mean? That the cold object must be sufficiently high density that it cannot be destroyed by the hot medium? Does the hot medium push on an initially under-pressurized cold object and force it to contract until it reaches equal pressure?

But wouldn't you expect hydrodynamical instabilities to destroy such cold objects as they move through the hot medium? For example, since the cold object is denser than the surrounding hot gas, wouldn't that density contrast cause a Rayleigh-Taylor instability and hence cause the cold cloud to sink? And then, since the cold object has some velocity wrt the hot gas, wouldn't Kelvin-Helmholtz instabilities shred the cloud? Under what conditions would these hydrodynamical instabilities be prevented so that pressure equilibrium is maintained and the cold object survives?

## 1 Answer

Pressure equilibrium means that the pressure of the hot component and the pressure of the cold component are equal at their interface, so that neither expands nor contracts relative to the other. The perfect gas law is $$P = n k T$$, so pressure $$P$$ is proportional to both density $$n$$ and temperature $$T$$. A cold cloud has low $$T$$, but if it has high density, it can have the same pressure as a hot, low-density cloud.

Implicit in this is that it is a more or less stable equilibrium, and it is relatively easy to see how this might be. Consider a cold (high-density) cloud embedded in a hot (low-density) medium. If a fluctuation causes the cloud to shrink, its density will increase, raising its pressure and causing it to expand. Or: if the temperature of the hot phase increases, so will its pressure, compressing the cold cloud -- until the latter's increased density (and possibly increased temperature) raises its pressure to point that it balances the hot medium's pressure again.

As for hydrodynamical instabilities: these are not universal, always-on phenomena. For a cold, dense cloud inside a hot medium, the interface will have an inward-pointing gravity vector (pointing toward the center of the dense, cold cloud). This means you have a low-density medium sitting "on top of" a high-density medium, which is the opposite of what's required for the Rayleigh-Taylor instability. (This is why the interface between the Earth's atmosphere and the ocean doesn't suffer Rayleigh-Taylor instabilities.) So: no Rayleigh-Taylor instability. (A cold cloud "sinking" in an external gravitational field is not an example of a Rayleigh-Taylor instability; it's a buoyancy effect, like raindrops falling down through an atmosphere.)

You also probably shouldn't assume that Kelvin-Helmholtz instabilities will instantly or efficiently "shred the cloud". Large cold clouds or streams may be orders of magnitude denser than the hot medium and can have significant self-gravity as well, which will tend to resist "shredding". (After all, the interaction of the solar wind and planetary atmospheres in the Solar System can generate Kelvin-Helmholtz instabilities, but Venus and the Earth -- not to mention Jupiter, etc. -- still have significant atmospheres.)