# 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

## 1 Answer

There are lots of studies considering Lagrangian dispersion. One critical aspect is the addition of random displacement to the deterministic particle track in order to consider subgrid-scale processes. Your measure of dispersion seems acceptable except for the fact that it might be dependent on the resolution of your flow field. In order to compare your dispersion with other areas or times, you might need to scale it by the horizontal diffusivity of the flow. Some studies that could be helpful are Mariano et al. (2002), Proehl et al. (2005), Lynch & Smith (2010).

• Thanks for the references. The Eulerian velocity field is calculated using a highly resolved DNS so I am thinking that a deterministic particle tracking method is fine? – Isopycnal Oscillation Aug 11 '14 at 16:58
• I assume deterministic is fine, but scaling might be still useful. – arkaia Aug 11 '14 at 17:42