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An infinite set of blank pages for our fearless leader. Created by John Gibson 2009-07-27 :-D

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2010-12-13 There are people who promise to US military to find long (relative) periodic orbits in boundary sheer flows by variational methods, for a significant amount of money. They presumably do not mean it, but just in case someone does: L-BFGS algorithm (one of quasi-Newton optimization methods) might be the method to implement.

2010-11-20 Probably should have a look at this paper: Schneider Tobias M.; De Lillo Filippo; Buehrle Juergen; Eckhardt Bruno; Dörnemann Tim; Dörnemann Kay; Freisleben Bernd Transient turbulence in plane Couette flow Phys. Rev. E 81, 015301 (2010)

2010-06-09 Paul Manneville and Joran Rolland: On modelling transitional turbulent flows using under-resolved direct numerical simulations: The case of plane Couette flow. While the dynamics of the transition from laminar flow to turbulence via localised spots can be investigated with reasonable computing resources in domains of limited extent, the study of the decay of turbulence in conditions approaching those in the laboratory requires consideration of domains so wide as to exclude the recourse to fully resolved simulations. Using Gibson's C++ code ChannelFlow, we scrutinize the effects of a controlled lowering of the numerical resolution on the decay of turbulence in plane Couette flow at a quantitative level. We show that the number of Chebyshev polynomials describing the cross-stream dependence can be drastically decreased while preserving all the qualitative features of the solution. In particular, the oblique turbulent band regime experimentally observed in the upper part of the transitional range is extremely robust. In terms of Reynolds numbers, the resolution lowering is seen to yield a regular downward shift of the upper and lower thresholds Rt and Rg where the bands appear and break down.

2010-06-09 Predrag to Ruslan, about a possibility of applying for BBSRC US Partnering Award scheme, an upcoming application deadline for November 2010:

Grigoriev, Fenton, Cherry and I applied for NSF cyber-initiative, idea being that the yet-unknown postdoc would work with Grigoriev, Cherry and me on finding numerically exact unstable relative equilibria (spiral waves & scrolls) and relative periodic orbits in excitable media. That would be a continuation of work of Biktashev(a), Barkley, and zillion others in our, turbulent direction, with hopefully leading to better diagnostic and control. So if you want to do that, we are biotechnology?

2010-01-06 Jean-Pierre Eckmann: Y R U doing all this?

2009-12-12 The thrilling SONG OF THE UNITED PLUMBERS UNION OF THE USSR, which I am pleased to forward to you!

2009-10-29, to John: can you make this page invisible to the vulgar public?

2009-10-29 Evangelos and I are learning a lot at Lyapunov analysis, from theory to geophysical applications workshop. I'm glad to hear that our pCf eigenvectors are Floquet. I have put Ginelli talk into so you can ponder whether you could get larger number of Floquet eigenvectors. But I'll continue this in the svn siminos/blog, as Ruslan and Evangelos do not read this particular rag.

2009-10-22 Let us now praise paranoia Thursday night might Dell XPS-M1210 laptop motherboard got fried, and has been as dead as a dead chicken ever since. The systems man Thomas who was running a virus scan on the windows partition from linux when it committed seppuku is so saddened by its demise that he will perform the autopsy the coming week.

It turns out that a modern man cannot leave without a laptop, especially if she/he is to give a talk Monday afternoon in Paris. Unlike YouKnowWho at Georgia Tech, people here jumped to help me on Friday, and loaned me a Toshiba PCtablet and a Dell Latitude D610. I was sadly and definitely defeated by the PCtablet with XPwindows in German, and keyboard in German, but luckily Bjorn's Dell laptop was bought in England. Still, it excuses one from thinking for two whole days to set up a laptop so it can be used. Here paranoia really pays off: (1) anything off substance is on the subversion server, and thus recoverable. (2) Every night my external Seagate hard disk backed up my system by FreeAgent Tools. (3) Most programs of import were installed on Seagate USB disk's Ceedo Windows partition, and can be run off any windows machine just by plugging in (still, they took lots of time to update). (4) the talk can always be given off

Except that I have a whole bunch of hand drawn Lyapunov-vector specific drawings that I was planning to include, but can forget now. Off to the train station and night train to Paris now. Hopefully by next week I'll learn how are we to compute our Floquet eigenvectors…

2009-10-01 I've now worked for several weeks around the clock in order to craft the chapter on continuous symmetries persuasive enough for Master Pipe Plumbers Marc and Ashley to try quotienting SO(2) from their pipe traveling waves and plotting them in our states space projections. They also have the Arnoldi-Krylov code to chart out unstable manifolds with. But all they do 8 hours a day is to tweak figures for the resubmission of their transient lifetimes JFM paper, I despair that I will ever get their ear, even bribing them with delicacies from the excellent Café Hemer behind the Institute. Duguet tried after I explained it to him in Kyoto, and then gave up for good. It should have not worked, as I had explained to him what we now call Rowley and Marsden's “method of connections,” but maybe I can get my brave plumbers to give a go to the “method of moving frames/slices.”

2009-09-29 JFG Don't bother. Short story is this: Wally wanted me to check out his hunch that transition from laminar to turbulence generically follows this path: a small perturbation from laminar follows the stable manifold of LB up towards LB and then gets kicked out to turbulence by LB's unstable manifold. I did some computations in a (4pi x 2 x 2pi) box, several times larger than HKW (roughly 2 pi x 2 x pi) and large enough to have LB eqbs at several spatial wavelengths. But I found no evidence that the transition occurred due to a close pass to any of them. As Divakar suggested, the scenario Wally outlined might well be right but with respect to some other solution's stable/unstable manifolds. I think this is a good project (and one that would make us even more famous), but we would need to check among a larger set of invariant objects for close passes . I think it would make most sense to do this in a small box where we know a bunch of solutions already, and with symmetry restrictions to lock phase. If that works, we could expand from there. Will copy this to the lengthscales blog as well.

2009-09-29, to John: Should I study the learned masters thoughts on scale and state selection? Or is in this case too ignorance bliss?

DSPDE's 2010

2009-11-30 Marc Barcelona May 31 – June 4 DSPDEs 2010 minisymposium proposal accepted, 4 time slots of 25'+5' minutes assigned for oral presentations.

Session Code: 46, Corresponding Organizer: Marc Avila Canellas, Title: Dynamical systems and the onset of turbulence in shear flows

Invited authors are expected to register (I registered as cvitanov), and send their abstracts. Find at the bottom “Add an abstract for a Minisymposium”. Fill in the correct Code Session. The “Topic” item can be ignored. The abstract (free format, 300 words approx, .pdf file) should describe the main features of the work.

  • January 15th., 2010: Deadline for presenting a one page abstract
  • February 15th., 2010: Acceptance letters will be sent for the presentations
  • February 28th., 2010: Deadline for payment of corresponding author’s registration fee.

Scholarships are available for pre-doc students

2009-07-30 On spatially localized solutions

PC 2009-07-30: Lan and I have done lots of work on spatially localized solutions and such for KS. I have not succeeded in getting him to write it up, and his wife did not order him to put it into his thesis, so all one can read is a bit in Sect. III of our KS paper, and in the Sect. “5.3 Steady solutions of the KSe” of Lan's thesis. Really pretty work was done by C.P. Dettmann when he visited me at Nortwestern, but he absolutely refused to write it up, so not a single picture can be shown today, at least not by me. The original good work that this would have been a continuation of was done by D. Michelson, Physica D 19, 89 (1986).

Everyone treats spatial extent as time; if you do it in both spanwise and streamwise directions, it will be cute, as you will have two (non-compact, in contrast to Lie group parameters) “time” parameters.

The beautiful pictures I cannot show you, as that was before the age of repositories and this blog, so they hae vanished into Lan black hole. But basic idea is that treating 1D PDE spatial extent as time turn KS into a 3-dimensional ODE dynamical system, with one “Energy” parameter. The dynamics of this system (ie, the totality of equilibria of the original PDE) is quite interesting, tricky, and elegant: it has its own pair of equilibria (I call them “equilibria of equilibria”), which organize all of the PDE equilibrium solutions, and explain spatial periods/lenghts typical of them. The set of all equilibria is an interesting fractal set in this 3- or 4-dimensional (if energy is viewed as 4th dimension) statespace. In principle, once this space is mapped out, there are no blind searches for equilibria, they have a symbolic dynamics of their own.

PDE restricted to a periodic domain has only a subset of spatially periodic solutions of given periodicity. While the number of unstable periodic solutions of the 4d dynamical system is infinite, the number of solutions of period shorter than Lx is always finite, so here one understands quite well why the number of PDE equilibria is finite, and can perhaps even bound it.

A systematic exploration of Navier-Stokes equilibria would replace the pCf equations with spanwise/streamwise “times” and a 2D ODE, whose equilibria and their (complex) stabilities would explain the spatial structures observed, most importantly the renge of wave numbers observed in the wave-number parametrized continuation/bifurcation plots. The reformulation would require a bit of thinking; the new equations have to be written down, and we have not dealt with two “time” axes before - maybe it is in the latest Marsden paper that John refers to in his blog, but I doubt it.

From the ergodic theory point of view there is no virtue in seeking “solitonic” or “spatially localized solutions.” That is related to the usual discussion of what is more effective in computing ergodic averages; heteroclinic orbits or periodic orbits?

From topological point of view, heteroclinic orbits are very nice, as they enable us to start with a few equilibria/periodic orbits, and use their unstable manifolds to partition the state space sharply and hierarchically.

From ergodic point of view, heteroclinic orbits are inconvenient, as they are infinite-time orbits and computationally not under control in higher dimensions. We also have no formulas for computing ergodic averages from heteroclinic orbits; I have succeeded in finding an exact trace formula for them only once (see formulas leading up to Table 27.2 chapter "Irrationally winding", but that was due to number-theoretic magic that I do not expect in any physics application.

Periodic orbits have a natural hierarchy - the most important ones are the shortest ones, the easiest to compute accurately. They probe the ergodic regions of state space in a uniform, systematic manner, with ergodic averages given by exact trace and determinant formulas.

Conceptually, both could be used - averaging is done by identifying characteristic structures of increasing characteristic time and spatial periods. For ergodic systems solutions that do a few repeats and then go heteroclinic to something quite different and those that repeat ad infinitum are exponentially close to each other within a single spatio-temporal “tile.” If the solution that they are heteroclinic to is marginally stable, “spatially localized solutions” might be quite physical, in the sense that one might observe large nearly laminar regions both in space and time, with intermittent bursts in-between. It is the question that we (as a community) should get together with spatiotemporal crowd in a workshop and try to sort out.

Good luck fishing.

Yeah, one of these days I'm going to get me a bowie knife and move up from fishing to huntin' 'coons. I can see the point of finding eqbs of KS or Swift-Hohenberg by converting the equilibrium equation to a 3 or 4d ODE where x is treated like time, and eqbs of the original PDE are periodic orbits of the 3 or 4d ODE. This is advantageous because the machinery (theoretical and computational) for dealing with low-d ODEs is well-developed. The utility of trying something similar to Navier-Stokes, where you might reduce the 3 space PDE system to a 10^N dimensional ODE, N in the range 3 or 4, is a whole lot less clear. JFG 2009-07-30 19:00 EST

2009-07-30: If I understand you correctly, you are saying localized solutions are uninteresting…

2009-08-04: Actually, not at all. What I do not understand about our equilibria is the wavenumber selection - as your continuations show, they exist for ranges of spanwise, streamwise periodicities. By looking at aspect ration → ∞ homoclinic trajectories which visit neighborhood of a spatially periodic equilibrium a number of times, as you do now, maybe you will explain the preferred wave number of spatial solutions.

If you find that some of these solutions keep hugging the wall and are converging with increasing Re to solutions robust when expressed in wall units, you would identify near-wall structures. Knock everybody's socks off!

If I understand you correctly, you are saying localized solutions are uninteresting because they would not play a role in trace formulae for ergodic averages over infinite spatial domains. This is because, in a related system in which the infinite spatial variables are treated as temporal variables, the localized solutions would be heteroclinic cycles (really, homoclinic, right?), whereas the spatially periodic equilibria would be periodic orbits, and trace formulae involve periodic orbits, not heteroclinic/homoclinic orbits. I don't understand a few things about this argument. (1) What is the relation between ergodic averages of the related system (i.e. the 3d ODE for KS you get by setting du/dt = 0) and ergodic averages of the original system? I can see how local equilibria (heteroclinic/homoclinic orbits) are irrelevant to ergodic averages for the *related* system, but I don't how spatially periodic equilibria (periodic orbits) have anything to do with ergodic averages for the original system. We need periodic orbits of the original system for that, and the objects under discussion are only its equilibria, localized and spatially periodic. (2) Lack of contribution to trace formula doesn't necessarily make a thing uninteresting.

The Marsden paper I refer to one that considers spatially exponential instabilities of plane Couette and pipe flow. I will find the reference.

2009-08-03: I understand “time-like” to mean this: for a differential equation with N independent variables x1, x2, .. xN, the variable xn can be taken as “time-like” if the equation can be written as du/dxn = f(u), where f involves differential operations in any independent variables but xn.1) A variable being time-like is advantageous because du/dxn = f(u) is a dynamical system; the evolution of u in the variable xn for xn>0 is completely specified by the value of u at xn=0, and generally, the mathematics and algorithms for analyzing such systems are understood.

2009-08-04: Agreed. For pCf it might be worth doing, picking, let us say, the most important, streamwise direction as “time” and rewriting 4D equilibrium (and streamwise traveling relative equilibrium) PDE condition as a 3D PDE with x (streamwise) “time” variable. Because of the Laplacian this is 2nd order in “time,” needs to be replaced by a pair of 1st order PDEs, etc. Have not tried, do not know if anyone has, do not know the level of programming difficulty in implementing 3D PDE. KSE turned out to be a nontrivial set of ODEs, so unstable that we needed Lan's steepest descent to nail the equilibria. The payback - if lucky - would be that the “equilibria of equilibria” would give us an explanation for the preferred spatial periodicities to expect in the infinite streamwise aspect ratio. Nothing to jump into with manpower we have now, but something to keep in mind.

2009-08-11: Agreed. Tobias and I have been talking about doing that for the localized EQBs for those reasons, so I guess I do see the utility in it. But first we will get a paper out this fall on the continuations, snaking, saddle-node bifurcations, and the relation of the local solutions to the spatially periodic solutions (the ones we have so far bifurcate from Nagata lower branch, once we understand that bifurcation we might be able to construct a zoo of local solutions from our periodic zoo).

2009-08-03: What does it mean for a system to have two time-like variables? Has anyone ever developed a trace formula for a system with two “time” variables? Or are you speculating?

2009-08-04: The “trivial” example I have is for flows equivariant under a compact Lie group of dimension N (parameterized by N angles: N=1 for SO(2), N=3 for SO(3), etc). Each “time” dimension is compact, each compact solution is characterized by one time period and N phase shifts, and as time evolution commutes with the Lie group symmetry, the trace formula is reduced to a sum over invariant subspaces of the Lie group.

2009-08-03: Presumably that the equation can be written du/dx1 = f(u) where f involves no differentials in x1, and as du/dx2 = g(u) where g involves no differentials in x2. The 1d wave equations du/dx = -c du/dt fits the bill. You can treat either x or t as time, that is, evolve u(x,t) for t>0 from the initial value u(x,0), or evolve u(x,t) for x>0 from the initial value u(0,t). What it would mean to treat the wave equation as a system with two time variables simultaneously, I don't know. Evolve u(x,t) for x>0, t>0 from initial condition u(0,0) (a single real number)? That clearly isn't going to work.

2009-08-04: I have no clue how two non-trivial non-commuting “times” characterizing spatially double-periodic (relative) equilibria and hetero/homoclinic solution of pCf could be integrated; my guess is that the only way to find them would be variational methods. That's what we already do for 1 “time” formulation of KS - “Newton descent” is a variational method, even though a referee forbade us to use that word.

Trieste 2009 Turbulence Conference

Turbulent Mixing and Beyond

2007-07-27 PC: Know about 4 people out of 160 attending

  • Nagata, for the duct flow:
    • Chebyshev x Chebychev x Fourier
    • corner cause numerical difficulties
    • Four symmetry invariant subspaces - found solutions within one (?)
    • has found his upper/lower branch pair
    • has found a solution pair “with inflection” by applying heat forcing, than taking it to zero
  • Our presentation, the first talk of the conference (so they were still awake)
    • used the version for the Attention Deficit Order challenged
    • lots of questions
    • an unknown (but already so old) Russian wants to know how does it compare to POD. I weasl politely around but essentially say that we do not use POD any longer, DNS is more useful for us.
    • Andrew J Majda goes off his rocker, as I did not refer to his work, which had already explained it all. In particular, he has explained that POD does not work because it is a long time average that misses rapid intermittent episodes described by his “heteroclinic chaos.” I say that I agree with him 100%. He leaves my talk, haven't seen him since. Later in the conference Rytis talked to him: Majda want us to read, refer to his paper with D. Crommelin, “Strategies for model reduction: comparing different optimal bases”, J. Atmos. Sci., Vol. 61, No. 17, 2206 (2004) on his publication list. (PDF here)
    • an unknown (but already so old) Russian states that for him 5 POD modes capture 99.8% of energy, never read any Gibson articles claiming that POD fails to do it with 1000.
    • Viktor L'vov started out as experimentalist, 15 years ago went for lunch and when he came back his Couette-Taylor experiment (one rotation per second) settled into a complicated but stable 2-hour periodic orbit. Read S.L. Lukashchuk, V.S. L'vov, A.A. Predtechenky and A.I. Chernykh. ``On the effective space of circular Couette flow and the structure of its attractors“, in ``Non-linear and Turbulent Processes in Physics”, ed. by

R.Z. Sagdeev, (Gordon and Brich Publ. NY 1984 pp. 1455-1464).

  • please read halcrow/TEX/main.pdf blog for initial description of Yakhot's suggestions
    • added suggestions from Yakhot and others;
    • asymptotic scaling might be obtainable from our solutions
    • read at least sect “2.2.2 PC 2009-07-27: Asymptotic exponents from low-Re” in the day's Bluesky Research section of the blog. All the papers referred to are in the
  • escaped to Istria later in the day huggin' my boy Rytis on the back of his motorcycle.

Photos, please! John Gibson 2009-07-27 12:00 EST

Sorry, no pics, but above is the link to the route — Predrag Cvitanovic 2009-07-29 19:08

Unstable periodic orbits - Tufts & University College of London

2009-11-29 JFG Bruce Boghosian invited me to give talk 2009-12-04 at 2 pm.

2009-11-15 PC Bruce Boghosian <bruce.boghosian [snail ]> contacted me, invited me to visit Tufts. He writes: “We met in Copenhagen in 1992. I was on an extended visit to the Technical University of Denmark, and I gave an informal talk on lattice Boltzmann models of fluids at the Niels Bohr Institute. I have worked on computational fluid dynamics for most of my career, with emphasis on lattice Boltzmann models of various sorts.

In 2005, at the International Conference on Discrete Simulations of Fluid Dynamics, I saw a fascinating talk by Shigeo Kida on finding periodic solutions of the driven Navier-Stokes equations in the turbulent regime. I recall that he commented that each solution that he presented took approximately 50,000 cpu-hours on the Earth Simulator machine in Japan, the fastest supercomputer in its day.

When I returned to the US, I began looking at some of the numerical challenges posed by this problem, first with an undergraduate student and later with a graduate student. We read the entire portion of your “Chaos Book” devoted to classical chaos, as well as the Cambridge monograph by Bohr, Jensen, Paladin and Vulpiani. Then my students and I went through the long process of catching up with the literature. As a result, I have become a convert of sorts to the dynamical systems approach to the problem of turbulence. I would very much like to ramp up my research activity in this area, applying my skills in computational fluid dynamics to the numerical challenges of locating unstable periodic orbits in turbulent flows.”

2009-06-30 JFG This page was world-readable, but I changed it to GTspring09-only. Agreed about Boghosian. I heard him talk about linear algebra algorithms on the Connection Machine when I was at los Alamos. I have the sinking feeling that Jonathan mentioned after reading Smeigel's thesis…. Lattice-Boltzmann is sub-optimal way to simulate Navier-Stokes, I think. Good for parellization, bad for discretization. I wonder if their solutions converge. This would seem to prove you right that is is computationally feasible to compute orbits variationally, in a spatio-temporally discretized search space. I should have been clearer in my earlier protestations that it was not feasible for me to do it in a time scale that either of us would be comfortable with.

2009-06-29 PC this info is confidential, so I am putting it on a password protected page - move it wherever else you see fit, as long as it is not world-readable. Here is a talk that will be presented in Trieste end of July (I remember Boghosian as a smart guy):

Author(s): L. Fazendeiro$^1$, B. M. Boghosian$^2$, P. V. Coveney$^1$, J. L{\“{a}}tt$^3$, S. Smith$^2$
Affiliation(s): $^1$ Centre for Computational Science, University College of London, 20 Gordon Street, London WC1H 0AJ, UK; $^2$ Department of Mathematics, Tufts University Bromfield-Pearson Hall, Medford, MA 02155, U.S.A.
$^3$ Institute of Mechanical Engineering, Ecole Polytechnique Federale de Lausanne CH-1015 Lausanne, Switzerland}
Position(s) of author(s)/presenter(s): PhD student (presenter/first author)

Abstract text: “We present a novel algorithm for the identification of periodic orbits in differential equations. With this new approach, both space and time are parallelized, and a search procedure minimizes a functional simultaneously towards the trajectory of the orbit in phase space and its period. We discuss in detail the methodology followed for the identification of unstable periodic orbits (UPOs) in the Navier-Stokes equations (NSE) for incompressible viscous fluid flow, simulated using the lattice Boltzmann method. It has been known for quite some time that driven dissipative systems exhibit finite-dimensional attractors which are replete with UPOs. The attractors can be thought of as the closure of the set of all such UPOs. The UPOs in turn provide a countable sequence of trajectories from which dynamical averages can be extracted using the zeta function formalism. Averages thus obtained are not stochastic in nature, i.e., they do not suffer from the problem of having the statistical error decay as the inverse square root of the number of decorrelation times. In this presentation we discuss the numerical difficulties involved in the numerical relaxion procedure for systems with very many dimensions such as the NSE, and describe the application of the conjugate gradient method to the problem. Since this method requires the storage in memory of spacetime trajectories, huge computational resources are required, which places the work firmly in the emerging field of petascale grid computing. We present results obtained on the IBM BlueGene/P at the Argonne National Laboratory and on the Sun Constellation Linux Cluster at the Texas Advanced Computing Centre, two of the world's current largest resources for open science. Results of turbulent flow simulations are presented and discussed and preliminary periodic orbits are presented. The insights that the identification and classification of these UPOs are expected to bring to turbulence theories are discussed.

2009-08-?? Slice & dice



On spatially localized solutions. Were I fearless I would get the dogged follower to sit down and write the famed POs paper. Purged, yes. Fearless my foot.

Initiated fearless leader blog. John Gibson 2009-07-27 12:03 EST

Might have allowed the independent variables to appear on the RHS, e.g. f(u,x1,..,xN), but that it not crucial.
gtspring2009/predrag/blog.txt · Last modified: 2010/12/12 17:49 by predrag