mgolden30/Kolmogorov_TW

Kolmogorov Flow Traveling Wave / Equilibria Database

Equilibria (EQ) and traveling waves (TW, also called relative equilibria) for 2D Kolmogorov flow at the parameters commonly used in the literature: nu=1/40, vorticity forcing -4cos(4y), periodic domain [0,2pi]^2, 128x128 resolution.

8,952 solutions (847 equilibria, 8,105 traveling waves), in a single file, solutions.h5.

Naming

Solutions are named by shape family, not by discovery order: F<family>-<n>, e.g. F17-3 is the 3rd solution in family 17. Families come from a symmetry-invariant shape clustering: recursive application of HDBSCAN (Campello, Moulavi & Sander 2013; sklearn.cluster.HDBSCAN, metric="precomputed", cluster_selection_method="leaf", min_cluster_size=10) over the full pairwise distance matrix, where distance between two solutions is their field L2 distance minimized over the problem's 16 discrete symmetries and continuous x-translation (a genuine metric -- every symmetry operation is an isometry, so this distance satisfies the triangle inequality). A single HDBSCAN pass leaves many solutions unclustered; we repeatedly recluster whatever's left over until a pass finds nothing further, which took 39 levels and produced 416 families, ranked F1..F416 by ascending mean number of unstable directions (F1 is the most dynamically stable family).

N-<n> solutions (658 of 8952, 7.3%) are genuine HDBSCAN noise -- the recursive clustering never placed them in any cluster, at any of the 39 levels. This is expected, normal HDBSCAN behavior (it is a density-based method that is supposed to leave outliers as noise, unlike e.g. k-means which forces every point into some cluster) and is not folded into any family: family == -1 for these, distinctly separate from family values 1..416. We do not claim N solutions belong anywhere -- do not treat them as a 417th "family," and do not assume two N solutions are similar to each other just because they share the N prefix; they are simply "not classified," each for its own reason.

(Historical note, for anyone cross-referencing a build from before 2026-09-19: an earlier version of this file folded those 658 noise solutions into the family of their nearest already-classified neighbor -- purely so every solution had a browsable name -- and labeled the resulting catch-all group U. That group's reported size, 725, was never a genuine HDBSCAN cluster size: it was 628 real cluster members (now family F253) plus 97 of the noise solutions glued on afterward for naming convenience. That scheme was confusing -- it made post-hoc-assigned noise look like real cluster membership -- and has been replaced by the honest N-<n> labeling described above. If you have data keyed by the old U-<n> names, cross-reference via legacy_name, which is unchanged by any of this.)

Within a family (not for N solutions -- see below), every member is rotated/reflected/shifted to minimize its distance to the family's own reference solution (member 1), so browsing a family actually shows one recurring shape, not 16 arbitrary orientations of it. The wave speed c is sign-corrected to stay consistent with this re-orientation. N-<n> solutions are not aligned to anything -- there is no family to align to -- and are stored in their raw, as-solved orientation. legacy_name records the original identifier (e.g. EQ7, TW118) from before any of this renaming, for anyone cross-referencing older material; it is stable across every naming scheme this project has ever used.

Fields (solutions.h5)

Field	Shape	Description
w	(8952,128,128)	vorticity field, symmetry-aligned
c	(8952,)	wave speed (0 for equilibria), sign-corrected to match w
x, y	(128,128)	full coordinate grids
forcing	(128,128)	-4cos(4y), same shape as w, ready to use directly
name	(8952,)	primary name, e.g. "F17-3" or "N-42"
family	(8952,)	-1 = N (genuine HDBSCAN noise), 1-416 = family number
solution	(8952,)	sub-index within its family
legacy_name	(8952,)	original name, e.g. "EQ7"
is_eq	(8952,)	True for equilibria
n_unstable	(8952,)	number of unstable directions (linear stability)
leading_growth	(8952,)	fastest growth rate, max(Re(log(lambda)))
injection	(8952,)	energy injection rate, mean(F.u) with F=(sin(4y),0)
dissipation	(8952,)	energy dissipation rate, nu*mean(omega^2)

injection and dissipation are equal to machine precision (~1e-15) for every solution here, as expected for a genuine steady/travelling-wave solution -- both are reported (rather than one implying the other) as a visible consistency check, matching e.g. Farazmand et al.'s convention of reporting both.

Root attributes: nu, Lx, Ly, axis_order (w's first axis is x).

More data: linear stability, DNS shadowing, and raw Schur vectors

This repository carries only solutions.h5 plus the two verification scripts. Three companion datasets that are too large or too specialized for this mirror are hosted at https://invariant-sets.org (see the KolmogorovRe40TW dataset page there for full field documentation):

• stability_observables.h5 (~330MB) -- per-solution linear-stability summary statistics (n_unstable, leading growth rate, the projected operator T, pairwise eigenvector angles), same row order as this file.
• stability_by_family/ (~38GB total, split into 417 per-family files) -- the raw Schur vectors (128x128 physical-space fields) underlying the summary above, so you can download just the families you need.
• turbulence_nn_search.h5 (~8MB) -- for every snapshot of a 6553.5-time-unit DNS trajectory of the same flow, the certified-nearest solution in this database under the full symmetry group.

Convention

Traveling waves are steady in a frame moving at speed c along x, defined via the standard traveling-wave ansatz w(x,y,t) = W(x-c*t, y) (as in f(x-ct) for any wave problem): positive c means translation in the +x direction. This is verified directly by tw_objective.py, which evaluates every solution against the governing equation with this exact sign convention and confirms convergence to ~1e-12 -- not just assumed.

Verifying the solutions

tw_objective.py evaluates every solution against the governing equation directly (spectrally, dealiased) and confirms convergence -- no MATLAB required. example.py shows basic usage and spot-checks a handful of solutions the same way. Both are plain Python (h5py, numpy) with no solver-specific dependencies.

python3 tw_objective.py solutions.h5    # checks all 8952, ~30s
python3 example.py                       # quick look + 5 spot checks

Every solution passes to a max residual of ~1e-12 (median ~2e-13).

Notes

• Multiple duplicate-detection passes have been run (checking field distance under the problem's full discrete+continuous symmetry group), removing solutions found to be near-machine-precision duplicates of another entry. No promises that there are absolutely none left.
• All solutions are converged below 5e-13 in the steady/TW residual as used during the search; this does not by itself mean the solution is physically accurate to that precision (no independent resolution/precision study has been done).
• Previous versions of this dataset shipped separate EQ.h5/TW.h5 files and MATLAB example scripts; both are retired in favor of this single file and pure-Python tooling.

License

CC-BY-4.0. Attribution: cite the associated paper, A catalog of exact steady states in Kolmogorov flow, or link back to this dataset.

Contact Matthew Golden (mgolden@lanl.gov) with questions or comments.