# 04.09.2014: Behavior of conserved-P system with chaotic advection and zero diffusion

Here, we study the behavior of the model using a conserved tracer (Ps in this case) initialized so that it is zero inside a disk and one outside. Biology has been turned off so that the only way the areas of 1 and 0 changes is because of numerical diffusion. Our goal is to be in a regime (strength of velocity field, intensity of stirring, model resolution) where the numerical diffusion is not too strong over several cycles (of the velocity field).

Several simulations are run for 50 days with increasing resolution (64, 128, 192 and 256 points). For each simulation, we plot first a series of snapshots and the histogram of Ps. If there was no numerical diffusion, the histogram would not change with time. With numerical diffusion, the initially bimodal histogram should become a single-peaked histogram.

With 64 points (Figs. 1 and 2), the domain is nearly mixed after about 40 days and the final value is around 0.8. Similar results are obtained with 128 points (Figs. 3 and 4) except that it seems the mixing is slower, which is expected. With 192 points (Figs. 5 and 6), results are very close to the previous case with 128 points. Finally, with 256 points (Figs. 7 and 8), the field seems to be rapidly mixed but the value plotted in Fig. 7 are not one but very large values (1e36, 1e54, etc) suggesting that there has been a numerical instability.

## Notes from meeting with Kelvin (04/14/2014)

- Check histograms
- Plot a cross section at day=20 or 25
- Read about the mixing down time in Richards
*et al*. (2006)
- One run with 512x512 points

Snapshots and histograms are plotted with `plot_snapshot_current_run.py` and `plot_diag_num_diff_3.py` in `RESEARCH/MODELISATION/marine_viruses/current_version/runs/run_002/analysis/` on `ipu1`.