The math is identical: potential vorticity conservation.

Particles carry vorticity, grid does the FFT Poisson solve, B-splines shuffle data between them. The trick is tracking how the flow map deforms (Jacobian evolution via RK4) and reinitializing before it blows up.

Why bother: grid methods smear turbulent structures. Lagrangian particles don't. Fusion people care because the scrape-off layer is where reactors fail. Turbulent blobs hit the walls.

Matches Arakawa finite-differences on energy/enstrophy conservation. Zonal flows emerge without forcing. Linear dispersion validates against theory.