The Boussinesq Wave Mechanism

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.