Conceptual Understanding of Tao’s Post

Derive the two-dimensional Boussinesq vorticity equation from its momentum equation.

Distinguish smooth preterminal evolution, forcing that remains smooth through the terminal time, and finite-time blowup of derivative norms in the Boussinesq theorem.

Verify that shared-phase temperature and vorticity waves do not advect themselves because of phase–velocity orthogonality.

Derive the wavevector transport equation on an affine incompressible background.

Derive the coupled temperature–vorticity amplitude equations by cancelling the leading Boussinesq residuals.

Diagonalize the frozen-coefficient amplitude system and calculate its exponentially growing mode.

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.