Spatially forced convection amplitude equation ($A_T=\mu A+\beta \overline{A}-\lambda |A{|}^{2}A$)

A steady spatial forcing at twice the critical wavenumber permits coupling of the negative critical Fourier mode to the positive critical mode. A weakly nonlinear expansion then permits a term $\beta \overline{A}$ in the Landau amplitude equation. Its coefficient is found by projecting the resonant forcing-advection terms onto the adjoint eigenfunction. Reflection-symmetric forcing permits real coefficients. The symmetry permission does not prove a nonzero coefficient: it can vanish for a particular model or mode structure.

For real $\mu ,\beta$ and $\lambda >0$, writing $A=x+iy$ gives a gradient flow with potential $V=-(\mu +\beta )x^2/2-(\mu -\beta )y^2/2+\lambda (x^2+y^2{)}^2/4$. The origin has exponential asymptotic stability for $\mu <-|\beta |$, is algebraically attracting at equality, and is unstable above it. Nonzero stable equilibria are the real pair $\pm \sqrt{(\mu +\beta )/\lambda }$ for $\beta >0$, or the imaginary pair $\pm i\sqrt{(\mu -\beta )/\lambda }$ for $\beta <0$, when their squared amplitudes are positive. Differentiating the two real equations gives eigenvalues $-2(\mu +\beta ),-2\beta$ on the real branch and $-2(\mu -\beta ),2\beta$ on the imaginary branch. For $\beta =0$, $\mu >0$ gives a radially attracting circle with neutral phase, so individual equilibrium points have Lyapunov stability but are not individually asymptotically attracting.

In the free-slip Stokes flow temperature model, let the critical temperature be $g=\sin \pi z$ and vertical velocity $f=sg$, $s=k_c^2+{\pi }^2$. A forced positive second harmonic $(W,\Theta )e^{2i{k}_{c}x}$ couples to the negative critical harmonic. Its temperature solvability condition has integrandThe identity follows by substituting $f=sg$ and differentiating. Its integral is zero because $g$ vanishes on both plates, regardless of the forced boundary value of $\Theta$. Thus the first resonant forcing coefficient vanishes. A generic nonzero conjugate-amplitude term cannot be inferred from wave-number matching alone.