# How to estimate dispersion in stratified flows

Consider a 2-dimensional domain with the typical x-z coordinates. Given an arbitrarily stratified flow, how can we estimate the depth varying horizontal dispersion of passive particles induced by the flow?

At the moment, I am seeding the domain with passive particles throughout the entire horizontal and vertical extent of the domain. Regarding the initial particle location, I employ a structured Gaussian distribution in the horizontal direction and a linear distribution in the vertical direction. Then, given the temporally variable Eulerian velocity field, I integrate all particle trajectories with a Lagrangian particle tracking code for the desired length of time.

To measure dispersion, first I calculate the Standard Deviation (sd) of the horizontal distance between the particles in the initial time ($sd(\Delta x_i)$) and the final time ($sd(\Delta x_f)$) and normalize it by its maximum absolute in the initial and final stage, respectively.

$$\overline{sd_1} = \frac{sd(\Delta x_i)}{max(abs(sd(\Delta x_i)))}$$ $$\overline{sd_2} = \frac{sd(\Delta x_f)}{max(abs(sd(\Delta x_f)))}$$

Given that, I use the difference between $sd_1$ and $sd_2$ as my depth variable dispersion measure.

Is there a better/more official way of doing it (taking into account that I want to use Lagrangian particle tracking methods).

• @aretxabaleta Deterministic, see comment to your answer below. – Isopycnal Oscillation Aug 11 '14 at 17:00