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2025-05-20 00:38:33 CEST

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It means that there are three invariants of motion for inviscid unforced fluids, sometimes called the Cauchy invariants (https://link.springer.com/article/10.1007/s00220-013-1848-1) associated with the local angular momentum of the fluid. In 2d, it means vorticity is conserved along fluid particles. In general, helicity conservation easily follows from vorticity conservation (eqtn 4.17 in the previously linked paper by Salmon).

2025-05-19 20:49:05 CEST

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In my opinion the deeper point is that vorticity conservation is a consequence of the gauge freedom of fluid mechanics: we can label particles however we want. See https://pordlabs.ucsd.edu/rsalmon/salmon.1988.pdf

2025-04-09 23:44:01 CEST

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Very fun read. While studying 2d turbulence on the ellipsoid, we used a nontrivial construction of ellipsoids based on the stereographic plane that might interest you: https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/twodimensional-turbulence-on-the-ellipsoid/05A6809F523F00109F08302ABCB4E54A

2024-12-26 04:50:29 CET

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They are not. There are a few ways to show this algebraically -- here's an image showing both (Bernoulli in blue, Euler in orange).
Clear["Global`*"]
\[Lambda] = 1; \[Mu] = -0.65223;
k = Sqrt[(1 - \[Mu])/2];
a = Sqrt[(2 (1 - \[Mu])/\[Lambda])];
z[s_] := a*JacobiCN[Sqrt[\[Lambda]]*s, k^2];
x[s_] := 2/Sqrt[\[Lambda]] JacobiEpsilon[s, k^2] - s;
A = 1.82;
ParametricPlot[{{A*Sin[s] Cos[s]/(1 + Sin[s]^2),
A*Cos[s]/(1 + Sin[s]^2)}, {x[s], z[s]}}, {s, 0, 4 \[Pi]}]
https://media.mathstodon.xyz/media_attachments/files/113/717/085/261/704/360/original/31a67ae5bc41181d.png

2024-12-24 23:23:25 CET

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Bernoulli's seems to be a subset of the Cassini ovals and are defined in terms of a simple algebraic relation. The so-called mechanical aspects of Euler's lemniscate shows the simple ODEs the curve satisfies: https://emis.de/proceedings/Varna/vol9/Djondjorov.pdf

2024-12-24 03:05:36 CET

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Here's a lemniscate from Euler that arises as a solution to the elastica (the shape a thin elastic rod takes when forced laterally -- these are curves that minimize the bending energy or equivalently the square of the curvature): https://math.stackexchange.com/questions/4509019/the-lemniscate-of-eulers-textitelastica-precise-determination-of-a-charac
see also: https://www2.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.pdf

2024-11-14 00:16:48 CET

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The climate science community has been migrating there and starting to thrive. It would be great to have your voice join some of these conversations, as I've learned a lot from you on these fronts over the years through your blog/twitter posts.