The Boussinesq Model and Wave Amplification

Interpret each term in the forced inviscid Boussinesq equations and identify transport, incompressibility, buoyancy, pressure, and external forcing.

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.

For a single plane-wave temperature–vorticity ansatz, derive the Fourier-mode Biot–Savart velocity and compute how the temperature gradient and velocity scale with frequency.

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.