From Controlled Layers to the Euler Paper

Analyze how controlled rotation resets a layer’s vorticity amplitude while retaining its temperature-gradient gain.

Explain how spatial cutoffs create compact nested affine cores and identify the localization residuals they introduce.

Construct a toy frequency-and-time schedule in which infinitely many layers fit before a finite terminal time while gradients diverge and field amplitudes remain bounded.

Derive the circulation and reduced-vorticity coupling for axisymmetric Euler flow with swirl in the Euler paper’s meridional variables.

Map the Boussinesq amplitude mechanism onto the Euler amplitude law, identifying the effects of material coordinates and spatially varying coefficients.