I was going through some stuff today and found these notes I typed up a couple years ago on the derivation of the “perturbation enstrophy budget” that I never posted! So I’m putting them here now in case anyone is stumbling around the internet looking for such things. I also wrote up some notes on the “perturbation potential enstrophy budget”, which is pretty interesting, but I can’t find them. Maybe in another year I’ll stumble on them too!

For those of you not familiar with what I’m talking about, vorticity is just a measure of rotation that is really useful in the study of fluid flows and wave dynamics. Enstrophy is a measure of “rotational energy”, which is analogous to how kinetic energy is related to velocity.

As a semantic side note, I tend to use the words “eddy” and “perturbation” interchangeably. Some people reserve “eddy” for describing when anomalies are defined relative to some zonal mean, whereas “perturbation” means anomalies from a temporal (i.e. time) mean. Personally, I think “perturbation” is a better term in in general, but neither of them should be restricted to a particular type of background state. I find is useful to consider perturbations relative to a background state that can still have variations in the dimension of interest. For example, if you want to define your perturbations with a high pass filter in time, then the low-frequency background state still has variations in time. This really affects how you derive a perturbation budget, although the notes here are a bit sloppy in this regard.

Anyway, here’s the link to the notes:

The Perturbation Vorticity and Enstrophy Budgets

Enjoy!