# Tank vorticity diagnostic, register v1

Classification: physically derived from the simulated MAC velocity snapshot; not an observation or an extra force. Intensive face velocities u, v, w have m/s units in right-handed coordinates x rightward, y downward, z into the tank. Vorticity ω = curl(u) has s⁻¹ units, with components (∂w/∂y − ∂v/∂z, ∂u/∂z − ∂w/∂x, ∂v/∂x − ∂u/∂y). Its magnitude is nonnegative. Positive ωz is clockwise when looking into the tank along +z; this viewing convention does not change when the camera orbits. Rigid rotation has vorticity twice its angular velocity.

This read-only process runs on requested display snapshots after pressure projection. First average each face component to cell centres, then use centred second-order differences internally and one-sided second-order differences at the first/last cell centres. These are resolved interior derivatives, not ghost-wall stress estimates. Float64 arithmetic produces components and magnitude; transferred display RGBA is Float32. The process cannot mutate velocity, salinity, dye, pressure, conservation ledgers or physical time. Turning the view on/off changes only diagnostic work and presentation.

Before implementation, literal fixtures require: zero and uniform translation give zero curl to 1e-12 s⁻¹; rigid rotation on each of three axes gives 2Ω to 1e-12, including one-sided boundaries; pure strain gives zero; doubling signed velocities doubles signed curl and doubles magnitude; invalid grid/array/spacing or nonfinite velocities must fail before committing output. A smooth sine shear has a second-order derivative error on 16³ and 32³ grids, with refined RMS error less than 0.3 times the coarse error. Computing and transferring a diagnostic must leave the canonical state bit-for-bit unchanged. Selected dense-drop snapshots must have nonzero curl and finite outputs; the new view does not establish convergence of turbulent morphology.

The strength palette uses a fixed 0–8 s⁻¹ display scale; the signed-z palette uses −8 to +8 s⁻¹ with an explicit viewing convention. Out-of-range values clip at labeled endpoints. Magnitude displays the vortex-ring envelope; signed z reveals opposite turning directions in a cross-section. Signed colors use a square-root stretch of normalized absolute vorticity about a dark neutral zero; the legend uses the same transfer. Opacity is an artistic exponential transfer of normalized rotational strength. Color, opacity and volume lighting are artistic, not measured water optics. No synthetic vortex lines, vorticity confinement or hidden perturbation is introduced.

Source: [MIT vorticity and strain-rate lecture](https://ocw.mit.edu/courses/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006/c1453542b4d710e22b7e30c9e6b7c755_f11.pdf).