Butterflies: notes for a post

[Updated: see end]

This is more in the nature of notes for a proper post, but I'll put it here. If you want to read some sense, check James Annan: still flapping. If you want some nonsense, then follow the link therein to RP :-)

But first: something completely different:



So, I took HadAM3 (64 bit version) and did two runs: one standard, and one where I perturbed the surface pressure at a single grid point (I forget where: somewhere in the Arctic I think) by a tiny amount (1e-10, or was it 1e-12?).

The graph below shows the growth in global-mean area-weighted RMS of the difference of the MSLP field between the two runs:



The x-axis is time in days (48 timesteps per day): 0-5 (top); then 0-15 (mid); then 0-89 (bottom). By day about 25, and definitely by day 60, the difference between the runs has saturated: their weather is totally different, so no further growth in RMS occurs. The 5-10 day oscillations in the last month are, I think, what you'd expect to see from weather evolution.

The y-axis is in Pascals. 100 Pascals is an hPa, ie 1 mb. Standard weather charts tend to plot pressure in contours of 4 hPa, so in real weather terms the diffs are sort-of negligible out to about day 15 (although this is a global value, so locally there will probably be bigger values).

There are clearly different phases in the difference growth. After day 25-ish there is a slow rise to saturation. From day about 10 to about 25 there is exponential growth. From day 2 to 10 ish there isn't much growth. And from the start to about day 1 there is another exponential phase.

If you look carefully, you'll see that the diff appears to be zero for the first few timesteps. This isn't really true. But the model output (as opposed to its internal variables) has been converted from 64 to 32 bit floats; and the difference is identically zero at 32 bit (but not 64).

Meanwhile, its interesting to look at the pattern of the diffs.



The top pic is about day 4 (in the not-much-happening phase). The middle, day 15 (in the exp growth). The bottom, day 31 (saturated). Note that the pics have a different contour interval. By days 15/31 we're into "real meteorology" and hence the MSLP field is most different in extratropics, as it always is (its tropical dynamics, folks). The fact that the biggest diffs on day 4 are in the tropics (is this convection being jigged?) says we're in a different regime, but I'm a bit unsure.

So there we are for now. Over to you, James.

[The original post was oct 15. I've updated the timestamp.

Update: I've been playing with the timestep-dependence of this. In the pic below, the black line (solid) is run "a" (std) minus run "b" (small pert); black dashed is a-c, where "c" is a different small pert. Blue is the same but for with all three runs done at 1/2 the timestep. Red also, but for 1/4 the timestep. The std timestep is half hour. There are plenty of caveats here: firstly, that this is really only playing. Secondly, that all I did was change the value marked timestep (actually, the value marked "steps per period" from 48 to 96 to 192) without checking that anything was going hideously wrong elsewhere. Thirdly, that if you compare this to the previous plot you'll notice that its spikier: because its 6-h data (instantaneous) not timestep data. Fourthly, that even if nothing is going hideously wrong, changing the timestep does give a different model (is the climatology the same? I don't know. Maybe). For example, the atmos is fourier-filtered at the poles for CFL reasons. A smaller timestep means less area if filtered.

Also: to answer JA's q: does the pert grow one-box-per-timestep (ie, unphysically?). Well no. It grows faster than that, because, ha ha, the perturbation is within the filtering zone so it grows into the entire northern polar cap within a few timesteps.

But: the plot below shows that the initial growth is *slower* with smaller timestep (well even this is complex: the 1/2 timestep run grows faster up to day 1; but then the "plateau" level reached by it is lower between days 2 and 10; and the 1/4 timestep plateau is lower again. Does this mean that a very very small timestep (arguably, as physical reality has?) would have a very low (zero?) plateau? I must try some even shorter timesteps. BUT then you end up in territory where the model was never designed for and it becomes rather dodgy.




Having 3 different runs at each timestep (more would be nice of course) allows you to see that during the day 15-30 growth phase, the timestep doesn't matter. But earlier, it clearly does.

Curious. I wonder what it means...

ps: sorry James. I'll get back to it :-(