# What is the difference between an inertial instability and a symmetric instability?

Inertial and symmetric instabilities are fluid instabilities present in atmosphere and ocean.

What are the difference between an inertial instability and a symmetric instability?

• I think they are the same, another name for it is centrifugal instability. – Isopycnal Oscillation Dec 9 '14 at 18:28
• @IsopycnalOscillation I think that in some context the symmetric instability may be include in a broader class of inertial like instability. Also I don't known if the symmetric baroclinic instability, conditional symmetric instability are sub class of symmetric instability. A lot of time symmetric instability look related to density field. The inertial instability is not. I guess that in most context inertial instability and centrifugal instability may be the same. – ucsky Dec 10 '14 at 8:13
• I think one needs to make it clear what is an instability. Broadly speaking, there are only two types of instabilities, global and local. In the former case, particles collaborate together globally to grow a wave of particular wavenumber that will eventually overturn into turbulent motion. The latter type of instability is more catastrophic, in the sense that a local region of the flow may be unstable, such as in gravitational instability, leading to rapid overturning and mixing. In this context, inertial instability is global and baroclinic instability is local. – Isopycnal Oscillation Dec 10 '14 at 19:11
• @IsopycnalOscillation baroclinic instability can be global too, see R. T. Pierrehumbert, 1984: Local and Global Baroclinic Instability of Zonally Varying Flow. J. Atmos. Sci., 41, 2141–2162 or The Global Baroclinic Instability in Accretion Disks. Hubert Klahr – ucsky Dec 13 '14 at 7:33

Symmetric instability is the case where a parcel is inertially stable to horizontal perturbations and statically stable to vertical perturbations but unstable to perturbations between the horizontal and vertical. An additional constraint is that the flow is symmetric and only varies in two dimensions. This instability was also first examined as centrifugal instability, but with added baroclinity. Conditional symmetric instability (CSI) is related to symmetric instability with the difference being that $\overline{\theta}^*_e$ rather than $\overline{\theta}$ surfaces are used to asses the instability.
• Please can you define what is $\overline{\theta}_e$ and $\overline{\theta}$. I'm not sure that I understand: you said that this instabilities are similar because we are both looking at the stability in horizontal plane but after you said that symmetric instability is unstable to perturbations between the horizontal and vertical meaning not in horizontal plane anymore. Thank you for the page numbers. – ucsky Dec 13 '14 at 8:52