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08.31.2013: First runs with spatial nonuniformity with and without diffusion

NOTA: The results below have the problem that a system that starts with zero values does not stay at in that state (see below how Ps goes abruptly to zero outside the disk). This is due to a small error in the code findifad.f. The runs have been performed again and the results are show in 10.22.2013: First runs with spatial nonuniformity with and without diffusion. Part 2.

Case #1

Case #1 is the case where we start with equilibrium values for the Ps/Pi/V system inside a disk, null population around and zero explicit diffusion. Although, as expected, nothing happens inside the disk (Figs. 1 and 2), the value of the Ps population unexpectedly and suddenly increases from 0 to 1 outside the disk. Numerical instability?

After talking with Kelvin, it does not seem that this due to a numerical instability. Outside the disk, the equilibrium (Ps=0,Pi=0,V=0) is unstable and the system may diverge to (1,0,0). I could probably check this using a zero-dimensional configuration. I should also look at the original Truscott and Brindley (1994)’s paper where they probably discussed the stability of (0,0,0).

../../../../../../_images/equilibrium_0_diffk_0_timeseries.png

Figure 1: Time series in case #1.

../../../../../../_images/equilibrium_0_diffk_0_snapshots.png

Figure 2: Snapshots in case #1.

Case #2

Case #2 is like case #1 except that there is diffusion (diffk=37.99 m2/day, a value that should be equivalent to the value used in Richards and Brentnal 2006; see param.m for more details).

In this case, without surprise, the disk is spreading (Figs. 3 and 4). Three oddities. First, the disk appears to spread faster along x and y, and that is probably due to a numerical artifact. Second, Ps is max while Pi and V are near zero at the edge of the disk but the oddity is in the fact that the outer bound of the edge spreads faster than the inner bound. We see this very well in Fig. 5 where the time series of Ps along x is plotted. Third, there is some artificial boundary effect with the edge spreading suddenly faster when it comes close to the boundaries of the domain (again, we see this well in Fig. 5).

After talking with Kelvin, the first oddity is not one. Because the domain is periodic in x and y, the gradients along x and y are larger than the gradients along the direction 45 deg. from x or y and that explains why the spread is different between these two directions. The second oddity is interesting. I need to see if this is the case also in the P only case, P-Z case and P-Z-V case. If the addition of V triggers a response different than, say, in the P or P-Z case, that would be potentially interesting. A third remark from Kelvin is that we could add a bit of mortality on Ps (even when Z is zero) to keep (0,0,0) stable.

../../../../../../_images/equilibrium_1_diffk_37_99_timeseries.png

Figure 3: Time series in case #2.

../../../../../../_images/equilibrium_1_diffk_37_99_snapshots.png

Figure 4: Snapshots in case #2.

../../../../../../_images/equilibrium_1_diffk_37_99_Ps_ts_fixed_x.png

Figure 5: Time series of Ps along x=50 km in case #2.


Snapshot and time series figures are plotted with plot_snapshot_current_run.py and plot_timeseries_current_run.py in RESEARCH/MODELISATION/marine_viruses/current_version/runs/run_002/analysis/ on ipu1.