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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.
