# 04.02.2014: Behavior of V P system with chaotic advection and zero diffusion – Part 3, High-resolution simulations¶

Here, I study the effect of numerical diffusion using 512x512 points. With the initial velocity field, numerical diffusion is small Fig. 1). Then, I changed the velocity field in order for the chaotic time scale to be much sorter (10 instead of 100 days). As we see, I need to increase significantly the amplitude of the flow to have any significant stirring (Figs. 2 to 4) but, when that stirring became significant, I obtained a lot of numerical diffusion (Fig. 4), which did not reduce with a higher number of iteration (IORD=6 instead of IORD=3) (Fig. 5).

## 100-day period and coef=100¶

If we take parameter a (maximum growth rate for Ps) as the rate of biological reaction, the Damkohler number is 0.3*100=30. If we take parameter gamma (rate of generation of viruses), the Damkohler number is 1.6*100=160.

Figure 1: Snapshots in case run_002/equilibrium_5_diffk_0_coef_1_highres/ run with IORD=3. The velocity field is shown in the lowest panels.

## 10-day period and coef=0.1¶

For this simulation and those following: If we take parameter a (maximum growth rate for Ps) as the rate of biological reaction, the Damkohler number is 0.3*10=3. If we take parameter gamma (rate of generation of viruses), the Damkohler number is 1.6*10=16.

Figure 2: Snapshots in case run_002/equilibrium_5_diffk_0_coef_2_highres/ run with IORD=3. The velocity field is shown in the lowest panels.

## 10-day period and coef=1.0¶

Figure 3: Snapshots in case run_002/equilibrium_5_diffk_0_coef_3_highres/ run with IORD=3. The velocity field is shown in the lowest panels.

## 10-day period and coef=1e4¶

Figure 4: Snapshots in case run_002/equilibrium_5_diffk_0_coef_4_highres/ run with IORD=3. The velocity field is shown in the lowest panels.

## 10-day period and coef=1e4 with IORD=6¶

Figure 5: Snapshots in case run_002/equilibrium_5_diffk_0_coef_4_highres_version2/ run with IORD=6. The velocity field is shown in the lowest panels.

## Notes from meeting with Kelvin (04/02/2014)¶

It is possible that the timescale of reaction of the biology is too fast. Proceed as followed: Use a lower number of points, set the reaction rate to zero and find a combination of period and flow amplitude that produces the type of stirring we are interested in. Then, slowly increase the reaction rate to different Da number (see Richards and Brentnall 2006).

## Notes from meeting with Kelvin (04/09/2014)¶

OK. A summary first of what has been done. Kelvin pointed to me Martinez and Richards (2010) in which they give properties of the velocity field we should use. In particular, the period varies from 4 to 20 days and the coefficient for the velocity field is 0.7 times the width of the box.

Next, I wanted to investigate the numerical diffusion on a system where only Ps is non-zero (equal 1) inside a disk stirred by the velocity field. What I found is shown in Fig. 6: The Ps field blows up quickly to 1 within a few cycles.

Figure 5: Spatially averaged Ps for the case where all quantities are initially zero except Ps equals 1 inside a disk. β is zero so that no viruses are generated and the velocity field is a stirring one with different period and intensity. Plotted with plot_diag_num_diff_2.py in RESEARCH/MODELISATION/marine_viruses/current_version/runs/run_002/analysis/ on ipu1. The simulations are: no_reaction_stirring_test_0, no_reaction_stirring_test_1, no_reaction_stirring_test_2, no_reaction_stirring_test_4, no_reaction_stirring_test_5 (model gives NaN values for no_reaction_stirring_test_3 which has 256 pts in x and y; see onerun_1.py in RESEARCH/MODELISATION/marine_viruses/current_version/runs/run_002/ in ipu1 for details.

Kelvin noted that outside the disk, the system is at an unstable equilibrium point and that there is some biology in the sense that as soon as there is a little of non-zero Ps due to numerical diffusion, this blows up quickly to 1.

So, here are the things to do:

1. Do not even use the biological component in the code. This way, Ps will be (should be) a conserved tracer.
2. Starts with 0 inside the disk and 1 outside and look at the probability distribution evolving with time. Because of the numerical diffusion, the pdf should evolve from bimodal to one peak in the middle.
3. Be sure that you use the right resolution so that numerical diffusion stays weak over 5-10 cycles at least.
4. Repeat 2 and 3 with Ps=1 and V=0 initially outside the disk and at their equilibirum values inside the disk. We should have the same evolution of that pdf than in the previous case.
5. Using this new configuration, find the minimum value of diffusion coefficient needed for explicity diffusion to have an effect (to be larger than the numerical diffusion). Then, study the system with increasing numerical diffusion.
6. Repeat 5 where Ps is initially 1 everywhere and there is a patch (disk) of small V in the center of the domain; study it increasing explicit diffusion starting with the minimum value.

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