What do you mean ? There is a wide range of vortices of different size. Are searching for a definition of coherent vortices?

In fact， up to now no.

Cristian Alejandro Reyna-Ramirez, the video you provided is difficult to understand. What is going on in your video?

My guess is that it has some sort of a jet interaction going on which creates vortices - in other words, it is an experimental fluid mechanics problem.

In experimental fluid mechanics problem we, first, measure velocity fields of the flow, second, calculate different characteristics of the vortices.

The characteristics of the vortices fall into two types: size and strength. Consider the size characteristic. Since a vortex does not have a clearly identifiable boundary, one can come up with different characteristic sizes of the vortex. The main mathematical formulas to calculate the size of a vortex are Gamma-criterion, lambda-criterion and Q-criterion.

The last two are more advanced ones, so let's focus on the Gamma-criterion. It is a rather basic one, easy to understand and easy to implement (I implemented it myself in Python). Here is the reference to the paper it was introduced in: L. Graftieaux, M. Michard, N. Grosjean, "Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows", Measurement Science and Technology, 12(2001), 1422-1429. You need equation (11) from that paper which gives you the Gamma2 value (Gamma2 is the size of the vortex, whereas Gamma1 given by eqn.9 gives you the strength of the vortex which you can use to identify the location of the center of the vortex).

Once you have found the size of the vortex, its strength and position, you can use either of these 3 characteristics (or a combination of them) to calculate the life time of the vortex. I don't know of any specific formula for it, but you can come up with your own approach. The first thing that comes to mind is to adopt the classic thing from the turbulence theory: find the correlation coefficient of the vortex strength at the given point over time and find the time scale of the vortex life by integrating the correlation coefficient over time. Another thing you could try is to write a Python code to track the size of your vortex over time and when it falls below some threshold, call it the time scale of the vortex life. I'm sure you can think of a number of other ways to find the time scale. Be creative.

The team behind VortexFitting Python package (https://github.com/guilindner/VortexFitting) did a good job implementing lambda-criterion and Q-criterion. Their package can track vortices and do other cool stuff. Recently, we have added support for VortexFitting in PIVPY Python package as well as added functions for Gamma1 and Gamma2 criterions (https://github.com/alexlib/pivpy). PIVPY is the package for the analysis of velocity fields obtained with the particle image velocimetry (PIV) technique. And you can't get away with finding vortices without measuring velocity fields.