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===== 2009-06-12 =====

I am starting work with Tobias Schneider on spatially localized solutions of plane Couette flow.
These solutions were found by Daniel Marinc in his master's thesis under Bruno and Tobias, with 
edge-state searches in several different large aspect ratio simulations. Interestingly, they seem
to unite with spatially periodic solutions under continuation in Re. My first step is to recalculate
these continuations and reproduce the connection. Below are a continuation diagram and several plots
of the localized equilibrium state at different Re. Note how the localized solution spreads out to the whole domain 
as Re goes from 200 to 160. The continuation is still running; I will update as more data comes in. 

{{:gtspring2009:gibson:local:globlocred.png?300}}{{:gtspring2009:gibson:local:globlocred2.png?300}}

(continuation figs updated 2009-07-8)


{{:gtspring2009:gibson:local:ulocre400.png?300}}{{:gtspring2009:gibson:local:ulocre192.png?300}}{{:gtspring2009:gibson:local:ulocre182.png?300}}{{:gtspring2009:gibson:local:ulocre171.png?300}}
{{:gtspring2009:gibson:local:ulocre162.png?300}}{{:gtspring2009:gibson:local:ulocre150.png?300}}

===== 2009-07-07 =====
 
Eigenvalues as function of Reynolds number for the spatially periodic solution, computed for (Lx, Lz) = (4 pi, 8 pi), i.e. four spanwise repeated copies of one spatially periodic state (the Re=150 plot above gives you the picture). The two plots below show the real part of the leading eigenvalues versus Reynolds number. Left: the 10 leading eigenvalues, sorted and colored by magnitude of Re λ. Right: The eigenvalues that cross the real axis, remixed so the color stays continuous as the eigenvalues cross. 

{{:gtspring2009:gibson:local:uglobbifirc2.png?400}}{{:gtspring2009:gibson:local:uglobbifirc.png?400}}


Note:
  * Many eigenvalues are repeated and/or complex, thus fewer than ten lines appear in the left plot.
  * The leading eigenvalue (black in the left plot) is isolated. 
  * The next eigenvalues (red/blue in the right plot) are doubled. 
  * In the regions where red and blue are distinct, each red line shows two real eigenvalues (same for blue).
  * Where blue overlaps red, we have two equal complex pairs. 
  * The doubling of eigenvalues corresponds to a half-cell phase shift in the eigenfunctions. (I am pretty sure of this but haven't confirmed it by inspection). 

The real eigenvalues crossing at Re=150 produce the spatially localized state shown in velocity-field plots above. The complex eigenvalues crossing at Re=143 should produce a traveling wave, standing wave, or periodic orbit. I am puzzled by the lack of further bifurcations at Re=150, since Tobias mentioned (I think) that many localized states bifurcate off at Re=150. But maybe you can produce many solutions around Re=150 by appropriate linear combinations of the independent phase-shifted eigenfunctions. Need to talk to Tobias about this and read up on simpler cases. // John Gibson 2009-07-07 16:03 EST//

----

Tobias Schneider 2009-07-07 

Hi John,

looks great!! We should definitely talk soon - I'll try to reach you tomorrow  -  today I was unfortunately so busy with talks and administrative stuff that I could not respond to your mail - sorry for that.

Brief comment on the expectation of many solution bifurcating from the non-localized one: What we observed is, that at Re=150.2 (btw. what is your stepsize in Re?) there are at least two solutions connecting to the periodic one. One is the localized slowly moving traveling wave that is the edge state at Re=400 (and has a single unstable direction at 400). This solution is invariant under s_z (see page 65 of Daniel's thesis for the exact definition). (it is an even solution as a function of spanwise coordinate when averaged in streamwise solution). The other solution is the localized fixed point that is invariant under s_{x,y,z}. Can you try to also follow this solution down in Re to check how it connects to the periodic solution?

That there should be more solutions bifurcation at the same point is probably not true. It would be the case, if we are looking at the 'lower' connection of the snaking curve with the periodic solution. However, I think we study the point where the snaking curve 'reconnects' to the periodic branch due to periodic boundary conditions. I guess, our domain is simply not wide enough to show a real turning around of the snaking curve. Would it be possible to take the localized solution (either the tw or the fixed point - the best would of course be both) at a point where they are well localized (say Re=300) and put them into an even wider box. To minimize interpolation errors in a box twice or if possible even 4 times as wide. Assuming that your code allows to detect saddle-node bifurcations and to follow them 'around the corner' we might get a full snaking that way.

Let's talk about that in more detail. Are you in town next week - I would really like to schedule a meeting both with you and John Burke - we could either meet at Harvard or also visit John at BU.

Best,
tobias


----
John Gibson 2009-07-09 

By the way, I am worried that the localized traveling wave solution
might be spurious. The residual at the given resolution is 1e-14 or
so, but I cannot get better than 1e-06 at any other resolution. E.g. I
transfer from Nx,Nz = 64,128 to 64,144, then the residual goes up to
1e-05, and I can't reduce through Newton-Krylov. The globally periodic
solution shows no such problem.
===== 2009-07-13 =====

Continuation of localized solution on larger domain. The results above are for a localized solution on a 4pi x 2 x 8pi domain. Note that as the localized lower-branch solution is continued down in Re, it starts to fill the whole box and merges into the globally periodic lower-branch solution around Re=150. Tobias suggested putting the localized solution in a larger box and seeing if it would then continue around a saddle-node bifurcation to give us a localized upper branch. Good news: it does! 


{{:gtspring2009:gibson:local:globlocred3.png?300}}{{:gtspring2009:gibson:local:globlocred4.png?300}}

These plots show dissipation versus Reynolds numbers for three equilibrium solutions: the "global eqb", a solution that is triply periodic in the (4pi,2,8pi) box (see Re=150 plot above),for which we have both upper and lower branches; the localized solution in (4pi,2,8pi) box (see Re=400 to 150 plots above); and the localized solution fit into a larger box (4pi,2,16pi) (shown below). I'm using a different definition of dissipation here because including the laminar dissipation and normalizing by volume obscures the fact the localized solutions in the two different boxes  are almost identical from Re=400 down to Re=190. The definition of dissipation here is 

<latex>
D' = \int_{walls} (\frac{\partial u}{\partial y} -1)
</latex>

compare to the usual 

<latex>
D = \frac{1}{Lx Lz} \int_{walls} \frac{\partial u}{\partial y}
</latex>

so D' = Lx Lz (D-1). 


The computation is pretty slow, because the grid is large 32 x 33 x 384 for (4pi,2, 16pi). This discretization was appropriate for the localized lower branch at Re=400 but looking at the spectra 48 x 33 x 256 would be more appropriate for solutions near the saddle-node. 


{{:gtspring2009:gibson:local:uloc2pi8pire400.png?300}}
{{:gtspring2009:gibson:local:uloc2pi8pire194.png?300}}
{{:gtspring2009:gibson:local:uloc2pi8pire172.png?300}}

----
skype chat: 
[12:11:50] tobias.schneider: Hi John - does the solution after the turn show an additional pair of streaks?
[12:12:57] John F. Gibson: Yes. The ones on the ends strengthen. There are plots on my blog now; annotation still coming.
[12:13:16] tobias.schneider: cooool!!! I really think that this is a major breakthrough - a saddle-node bifurcation that gives an additional pair of streaks!!! We have indead a snaking bifurcation!
...
[12:19:56] tobias.schneider: perfect - if you have enough time to do that - could you try to also plot the spectra at the saddle node bifurcation point - that would be great
[12:42:20] John F. Gibson: I will try to do that by the time we meet John Burke. These computations are slow!
===== 2009-07-14 =====

I found  another localized solution by a simple modification of the existing solution (examined above). 
Let EQa be the existing solution. It's pictured below at Re=173.25. In the interior there's a pattern 
that repeats itself. I spliced out one copy of that pattern to produce EQbguess2 and then ran Newton-Krylov 
hookstep to get the new solution EQb. The picture of EQb is shifted a little to the right in z in order to
center it, approximately. 

{{:gtspring2009:gibson:local:eqa.png?400}}

{{:gtspring2009:gibson:local:eqbguess.png?400}}

{{:gtspring2009:gibson:local:eqb.png?400}}

{{:gtspring2009:gibson:local:eqcguess.png?400}}


I started a continuation of EQb to add to the D' versus Re plots. Also, it looks like one pattern could be spliced 
out of EQb as well, so I did that (EQcguess) and have a search running on that. I did a crude job on splicing, merely copying gridpoint values of velocity from one field to another, and taking a jump of one pattern length (plus or minus 
the gridspacing Delta z) in the middle. This messes up the continuity of the solution, induces Gibbs phenomenon, and gives non-zero divergence. Rather than spend time doing a better job I just time-integrated the crude guess for few time units until the Fourier spectrum adjusted (killed high-order modes). This zeroes the divergence as well. It seems to be good enough. 

**Update: Oops!** EQb is the same as the lower branch of EQa. So no new solution. I guess this is how snaking bifurcation works.... you go around a saddle-node and gain/lose one copy of the pattern. I am continuing with the EQc guess because it   would have one less copy than EQa's lower branch and so presumably be altogether different, if it converges without changing too much. We'll see. 

**Update 2009-07-15** EQcguess converges to the lower branch of EQa as well. Its tails grow under Newton-Krylov iteration.
===== 2009-07-15 =====

Updated continuation plots. The Back to the usual definition of dissipation, including laminar and normalized by box volume. This will be better to show the snaking connections to the globally periodic solution.

{{:gtspring2009:gibson:local:globlocred7.png?400}}{{:gtspring2009:gibson:local:globlocred8.png?400}}

These fields are points on the green curve. All are for 4pi,2,16pi. 

{{:gtspring2009:gibson:local:re_179.98_d_1.28.png?400}}

{{:gtspring2009:gibson:local:re_172.24_d_1.35.png?400}} 

{{:gtspring2009:gibson:local:re_172.31_d_1.40.png?400|foo}}

{{:gtspring2009:gibson:local:re_175.35_d_1.46.png?400}}

----

**Resolution** These computations are expensive enough that it's worthwhile finding the minimum acceptable resolution.
I recomputed the Re=174.36, D=1.484 4pi x 2 x 16pi localized solution at variety of resolutions. Spectra and properties of solutions are shown below.

{{:gtspring2009:gibson:local:spectra0_32_33_192.png?300}} {{:gtspring2009:gibson:local:spectra0_32_33_256.png?300}} {{:gtspring2009:gibson:local:spectra0_32_33_384.png?300}} {{:gtspring2009:gibson:local:spectra0_32_49_256.png?300}} 
{{:gtspring2009:gibson:local:spectra0_48_33_256.png?300}}  {{:gtspring2009:gibson:local:spectra0_48_33_384.png?300}}

The quantity shown in the Fourier spectrum is 

<latex>
\surd \sum_{i=0,1,2} \int_y \hat{u}_{kx,kz,i}^2(y) dy
</latex>

In my experience six orders of magnitude between largest and smallest Fourier coefficients is a pretty solid guarantee of stability of solutions under increasing resolution. The Chebyshev spectrum shown is

<latex>
\sum_{i=0,1,2} |\hat{u}_{kx,ky,kz,i}|
</latex>

as a function of ky for a few of the lowest-order values of (kx,kz).

|  grid  |  D  |  init  |  final  |  feval  |  minutes  |
| 24 33 128 | 1.505 | 3e-04 | 3e-05 | 216 | 059 |
| 32 33 192 | 1.48302 | 2e-05 | 2e-15 | 149 | 067 |
| 32 33 256 | 1.48401 | 4e-06 | 2e-15 | 108 | 066 |
| 32 33 384 | 1.48393 | 4e-06 | 1e-13 | 129 | 114 |
| 32 49 256 | 1.48401 | 4e-06 | 2e-15 | 112 | 122 |
| 48 33 256 | 1.48397 | 8e-06 | 4e-15 | 112 | 174 |
| 48 33 384 | 1.48390 | 8e-06 | 2e-14 | 099 | 231 |

32 x 33 x 256 for 4pi x 2 x 16pi seems to be the sweet spot. To get a few rungs on the snaking ladder, I am starting continuations of the localized solution on boxes with Lz = 24pi and 32pi, with Nz = 384 and 512 respectively.
===== 2009-07-16 =====

A few more rungs on the ladder in the 4pi x 16pi box (green) and a few points on 4pi x 24 pi (magenta)
and 4pi x 32pi (cyan). I've marked a few spots on the 4pi x 16pi curve that show corresponding points 
on successive rungs of the ladder. Midplane velocity fields for those solutions are shown below. Tobias,
John, please add comments to remind me what we are looking for between successive rungs.

{{:gtspring2009:gibson:local:globlocred9.png?400}}

{{:gtspring2009:gibson:local:loc2pi8pid1.519.png?400}}

{{:gtspring2009:gibson:local:loc2pi8pid1.445.png?400}}

{{:gtspring2009:gibson:local:loc2pi8pid1.358.png?400}}

{{:gtspring2009:gibson:local:loc2pi8pid1.306.png?400}} **Warning: all these should be labeled 4pi x 16 pi!**


BTW, I've confirmed that the globally periodic solution is the same as the Nagata lower branch EQB, by continuing down one copy from 4pi x 2pi to 2pi/1.14 x 2pi/2.5, for which I have the Nagata EQB.

**Yikes! I've been off by a factor of two on my box sizes through most of this discussion! I went back and changed all the values in the text but some figures have box sizes in labels. I won't change those; it's too much trouble. Box sizes in figures above this point should be doubled, below this line they are correct.**

Also, the only symmetry satisfied by this solution is s_xyz, where s_xyz : [u,v,w](x,y,z) -> [-u,-v,-w](-x,-y,-z), with a suitably chosen origin. I've checked in detail. The globally periodic solutions have more symmetries, due to the possibility of combining phase shifts with s_z : [u,v,w](x,y,z) -> [u,v,-w](x,y,-z) and s_xy : [u,v,w](x,y,z) -> [-u,-v,w](-x,-y,z). 
===== 2009-07-18 =====

Further progress on continuing solutions in 4pi,16pi and 4pi,24pi.

{{:gtspring2009:gibson:local:globlocred10.png?300}} {{:gtspring2009:gibson:local:globlocred11.png?300}}

Some comments on computation. After determining the symmetries of the solutions I optimized the phase shifts in x,z so that the center of symmetry is exactly (0,0) or equivalently (Lx/2, Lz/2) and also made sure the solutions were in the same orientation about the center. We had a brief power outage last night that caused my machine to reboot; this gave a good opportunity to restart the computations with the optimized phase shift, with sxyz symmetry enforced during time integration and continuation, and with the continuation program set to give up and try a smaller continuation step if the Newton-search steps are poor enough that the trust-region hookstep mechanism kicks in. Observations suggest that continuation is more efficient in total computation time if we never leave the convergence region of pure Newton iteration. Also, I reduced the discretization of the 4pi,16pi continuation to 32 x 33 x 256.

I started Arnoldi iteration on the phase-shift optimized versions the previously calculated solutions for 4pi,16pi. I didn't want to start these without symmetry optimization, since I need to examine symmetry to sort the eigenvalues, and it's a lot easier to detect the symmetry of the eigenfunction if you know the center of symmetry of the solution. That's running now. Looks like about an hour per solution at 100 Arnoldi iterations per solution. With two CPUs going and about fifty solutions (every other dot on the green curve), it'll take 24 hrs or so. It's clear from the first calculation that 100 Arnoldi iterations is not overkill. The first calc is at the 66th iteration, and the second zero eigenvalue is not yet resolved well enough to distinguish it clearly from the small non-zero eigenvalues. Arnoldi iteration generally requires more iterations to get accurate eigenvalues than GMRES does to get an sufficiently accurate solution. No great accuracy is required in GMRES solutions to Newton steps, since the Newton procedure is iterative itself. 
===== 2009-07-20 =====

Further progress on continuing the localized solutions 4pi,16pi and 4pi,24pi. 4pi,16pi appears to be headed to the upper branch of the Nagata solution. 
The Arnoldi iterations are still cooking. I had to restart them with a longer integration time (T=20) for the finite-time map u-f^T(u) in order to get better separation of the eigenvalues near zero. I've set up a bash script to do Arnoldi iteration on every other data point in the 4pi,16pi continuation; those should finish today, and then maybe I'll start it on the intervening points. Following that I'll have to solve the bookkeeping problem to track individual eigenvalues between continuation steps. In the meantime I'm going to work on some Poincare sections that have been stalled for some time...

BTW, the limiting factor on these computations is going to be memory in the Arnoldi iteration. The algorithm requires an LU decomp of an M x N matrix, where M is the number of independent spectral coefficients in the discretization of the velocity field, and N is the number of Arnoldi iterations. For the 4pi,16pi box with 32 x 33 x 256 gridpoints, that's a 200,000 x 100 matrix, and the process size gets to 1.2 GB during the LU decomp. My machine has 4 GB DDR3 1600 RAM and 4 cores, so I can do a couple Arnoldis at a time without trouble. But I expect to hit my upper limit pretty soon (a factor of 2 or 4) in domain size or Reynolds, or even sooner with less smooth solutions, which will need finer resolution. The Nagata LB is unusually smooth. I could put in two more 2 GB sticks to move up to 8 GB RAM, but then I'd have to downclock the RAM to 1033. I think the PACE machines are all 4 GB. 

{{:gtspring2009:gibson:local:globlocred12.png?300}}
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