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For $y^{\prime \prime }+y=\epsilon (y^{\prime }{)}^3$ with positive $\epsilon$, the oscillator energy increases at rate $\epsilon (y^{\prime }{)}^4$. The amplitude-phase equations give $R(T)=R_0(1-3R_0^{2}T/4{)}^{-1/2}$. The weakly nonlinear expansion fails when the growing amplitude makes $\epsilon R^2$ large, so its formal finite-time divergence does not itself control the exact late-time trajectory.

For $A_T=-\overline{A}B/2$ and $B_T=-A^2/4$, write $A=R{e}^{i\theta }$, $B=P{e}^{i\varphi }$ and $\delta =\varphi -2\theta$. The polar equations areWhere the polar phases are defined, direct differentiation gives the first integrals $R^2-2P^2$ and $R^{2}P\sin \delta$. In particular $R^2{\theta }_T$ and $P^2{\varphi }_T$ are constant. If $\delta =\pi$ and $R^2-2P^2=C>0$, then $P_T=P^2/2+C/4$, a Riccati equation whose positive solution develops a finite-time pole. This is finite-time blowup of the reduced amplitude system; it does not establish blowup of the full wave equation beyond the domain of the weakly nonlinear expansion.

Harmonic balance substitutes a finite Fourier series into a differential equation and projects the residual onto the retained harmonics. It determines their amplitudes and phases. A nonlinear equation generally generates additional harmonics, so vanishing of the projected residual alone does not make the finite series an exact solution. In a weakly nonlinear expansion, nonresonant generated harmonics can be found at the next order by solving their linear equations.

For the second-harmonic feedback in a long-wave convection amplitude equation, at $\mu =2$ the linear steady operator is $L_0=-(1+{\partial }_x^2{)}^2$. Its eigenvalue on $e^{inx}$ is $-(1-n^2{)}^2$, so the critical Fourier modes are $n=\pm 1$. More generally the linear dispersion relation is $\lambda (k)=-1+\mu k^2-k^4$, giving first onset at $\mu =2,k=1$.

At order ${ϵ}^2$, the weakly nonlinear expansion givesWith ${\Theta }_1=A{e}^{ix}+\overline{A}{e}^{-ix}$,The constant part disappears under differentiation. Inverting $L_0$ on the second harmonic, whose eigenvalue is $-9$, givesup to a critical-harmonic correction absorbed into the definition of $A$. At order ${ϵ}^3$,The coefficient of $e^{ix}$ in the last three terms is respectivelyThe Fredholm solvability condition requires their sum to vanish, because $L_0$ annihilates the critical Fourier mode. For a nonzero amplitude,This requires ${\mu }_2/(3-8s^2/9)>0$. The cubic amplitude coefficient is positive for $s^2<27/8$, yielding a small-amplitude branch in a supercritical bifurcation; it is negative for $s^2>27/8$, yielding a subcritical bifurcation branch in this leading approximation. At $s^2=27/8$, the cubic coefficient vanishes and the quoted relation is singular: higher-order nonlinear terms and a different detuning balance are needed. One must not assert a finite $p$ there.

If a term $ϵ{\mu }_1$ were included in $\mu$, order ${ϵ}^2$ would also contain $-{\mu }_1{\Theta }_{1xx}={\mu }_1{\Theta }_1$. The quadratic nonlinearity produces only the zeroth and second harmonics, with the zeroth differentiated away, so it cannot balance a first-harmonic contribution at that order. Solvability would give ${\mu }_{1}A=0$. Hence a nontrivial critical-mode expansion forces ${\mu }_1=0$ and first balances detuning against cubic amplitude effects at order ${ϵ}^3$.

The unforced onset is a stationary pattern-forming instability. Under a horizontal translation, the critical Fourier mode transforms as $A\mapsto e^{i\varphi }A$. A cubic amplitude equation without forcing must have the same phase weight: its leading terms are $\tilde{\mu }A-c|A{|}^{2}A$. Reflection of the unforced spatial pattern conjugates $A$, permitting real coefficients in this stationary problem. They are determined by a weakly nonlinear expansion and projection onto the adjoint eigenfunction; symmetry alone does not calculate their values or guarantee a nonzero coupling.

Represent the third-harmonic forcing by a complex coefficient $F$ multiplying $e^{3i{k}_{c}x}$, whose phase weight is three. The product $F{\overline{A}}^2$ has weight $3-2=1$ and therefore resonates with the critical positive harmonic. Neither a direct third-harmonic term nor $F\overline{A}$ has the required wavenumber balance. A travelling boundary pattern makesA response phase and the sign of its coefficient can be incorporated into $\delta$. The resulting three-to-one spatially forced amplitude equation isThis retains the leading resonant forcing term, linear detuning and cubic saturation, while dropping higher powers and nonresonant harmonics. The forcing is weak, the unforced critical eigenvalue is near zero, and the amplitude varies on a slow time; very high forcing frequency outside that slow scaling would require a different averaging argument. Reduction to the specific saturating canonical form in part (i) additionally assumes $c>0$ and nonzero forcing.

Rotating Rayleigh-Bénard convection combines buoyancy-driven instability with the Coriolis force. Consider a plane layer of depth $d$ rotating uniformly about the vertical axis, heated from below. Adopt the Boussinesq approximation, fixed boundary temperatures and, for explicit formulas, impermeable stress-free boundary conditions. The conductive state of Rayleigh-Bénard convection is motionless with a linear temperature profile. The dimensionless controls are the Rayleigh number, Prandtl number and Taylor number:Here $\nu$ is kinematic viscosity and $\kappa$ thermal diffusivity. In thermal-diffusion time units, linear perturbations satisfyRotation does no direct mechanical work, since $\mathbf{u}\cdot ({\mathbf{e}}_z\times \mathbf{u})=0$, but couples vertical motion to vertical vorticity and changes the damping and oscillation balance.

For horizontal wavenumber $k$ and vertical mode $n$, put $m=n^2{\pi }^2$ and $a=k^2+m$. Use $w=W\sin (n\pi z)$, $\theta =\Theta \sin (n\pi z)$ and vertical vorticity $\zeta =Z\cos (n\pi z)$ with growth rate $\sigma$. Curling the momentum equation and eliminating pressure givesTheir determinant, without dividing by a possibly zero factor, is the rotating-convection growth-rate polynomialThis makes the linear stability analysis question precise: onset occurs when a root reaches zero real part and all other modes still decay.

A stationary neutral root has $\sigma =0$, givingRotation raises this stationary threshold. An oscillatory neutral root has $\sigma =i\varpi$ with $\varpi \ne 0$. Real-imaginary separation gives the oscillatory neutral curve of rotating convectionThis branch is admissible only when ${\varpi }^2>0$, requiring $P<1$ and sufficiently strong rotation. The restoring Coriolis force coupling permits an inertial/thermal oscillation whose phase-lagged buoyancy can overcome dissipation. At large $P$, temperature and momentum diffusion do not permit that overstability mechanism at primary onset, so the exchange of stabilities is stationary. The actual threshold is the minimum of the stationary and admissible oscillatory curves over all allowed modes, not an arbitrary formal value of $R_o$. Rigid plates require a different vertical eigenproblem and Ekman layers, so the explicit free-slip numbers are not universal.

At large Taylor number, the first vertical mode is selected in the ideal plane layer. Let $t=k^2,m={\pi }^2$. Minimizing the stationary curve givesThus the stationary neutral curve of rotating convection hasThe physical horizontal wavelength is $2\pi d/k_c$, hence decreases as ${\mathrm{Ta}}^{-1/6}$; its prefactor depends on the boundary convention. Thin nearly vertical cells reconcile the strong Coriolis force constraint with viscous and thermal diffusion. The oscillatory minimization replaces the right side of the wavenumber equation by $P^2\mathrm{Ta}m/(1+P{)}^2$, giving the same Taylor number exponent at fixed positive $P$. Where its frequency remains admissible,Equality is $8P^4-P-1=0$, whose positive root is approximately $P_{\ast }=0.6766$. Accordingly, for sufficiently rapid rotation in this free-slip problem, $P<P_{\ast }$ selects oscillatory onset and $P>P_{\ast }$ stationary onset. The weaker condition $P<1$ is only necessary for an oscillatory neutral mode; it does not by itself identify the first instability. Finite Taylor number, finite lateral geometry, allowed discrete wave numbers and plate conditions change the selection.

For the counterpropagating Hopf amplitudes in rotating convection near a simple oscillatory onset, the Hopf bifurcation produces slow complex amplitudes for counterpropagating roll waves. After separating the fast carrier oscillation, symmetry permits the cubic equationswith generally complex coefficients; an $i{\omega }_0{Z}_{\pm }$ term restores the fast frequency if desired. The real parts govern amplitude saturation and the imaginary parts give nonlinear frequency shifts. Write $a_s=\mathrm{Re}{g}_s$, $a_c=\mathrm{Re}{g}_c$. For a travelling wave from a supercritical bifurcation with only one amplitude nonzero, $|Z_{\pm }{|}^2=r/a_s$ with $r,a_s>0$, and the competing wave's growth rate is $r(1-a_c/a_s)$. It is amplitude-stable against that competitor when $a_c>a_s$. A standing wave has equal intensities $r/(a_s+a_c)$; provided this is positive, its intensity-difference mode is stable when $a_s>a_c$. Temporal and spatial phase symmetries leave neutral phase directions, so these are orbital/amplitude stability statements, not decay of every phase displacement.

These coefficients follow from nonlinear interactions and the Fredholm solvability condition obtained by projection onto an adjoint eigenfunction; symmetry alone cannot decide their signs. A negative saturating coefficient gives subcritical bifurcation behavior requiring higher-order terms. Spatial modulation leads to coupled complex Ginzburg–Landau equations with group velocities and diffusion; phase instabilities, mean-flow coupling and differently oriented rolls can destabilize a wave stable in the restricted two-amplitude system. A weakly nonlinear expansion of oscillations therefore predicts travelling waves or standing waves, frequency shifts, modulation and possible secondary mode competition, not a unique universal periodic state.

The Küppers–Lortz instability is a different route to time dependence: it destabilizes steady saturated rolls against oblique roll perturbations. For stationary-roll amplitudes of orientations ${\varphi }_j$, a leading competition system hasA pure roll has $|A_i{|}^2=r/g_0$ with $g_0>0$. An infinitesimal new roll at relative angle $\vartheta$ grows atFor sufficiently strong rotation in appropriate boundary and Prandtl number regimes, some finite oblique angle has $g(\vartheta )<g_0$, so a steady roll is unstable arbitrarily close above its stationary onset. Rotation is handed and allows $g(\vartheta )\ne g(-\vartheta )$, so replacement of one roll by another can favor a definite cyclic sense. Three or more competing orientations can form a heteroclinic cycle; whether it attracts depends on contraction/expansion rates and other modes. Noise, spatially varying domains and modulation can turn this competition into repeated orientation switching and irregular patterns.

The invading rolls are three-dimensional disturbances even when the original straight roll is described by a two-dimensional section. The finite-angle Küppers–Lortz instability mechanism should also be distinguished from the small-angle instability of rotating convection rolls mediated by large-scale mean flow at finite Prandtl number. Numerical thresholds and favored angles depend on mechanical boundaries and material parameters; the essential criterion is the cross-coupling relative to self-saturation. Rotation both changes primary onset and wavelength, and can prevent the resulting steady roll pattern from remaining a stable nonlinear state.

Eliminate the instantaneous Stokes flow velocity in favour of temperature. On a horizontal Fourier mode $k$, the Stokes temperature-slaving operator maps $\theta$ to $w={\mathcal{W}}_R\theta$, where $(D^2-k^2{)}^{2}w=R{k}^2\theta$ and $w=w^{\prime \prime }=0$. The temperature evolution has linear operator ${\mathcal{L}}_R=D^2-k^2+{\mathcal{W}}_R$ and bilinear map $\mathcal{N}(\varphi ,\chi )=-{\mathbf{u}}_R[\varphi ]\cdot \nabla \chi$. Under the homogeneous thermal Dirichlet boundary conditions, ${\mathcal{L}}_{R_c}$ is self-adjoint. Normalize its critical eigenfunction as $g(z)=\sin \pi z$ and set $s=k_c^2+{\pi }^2$; the critical vertical velocity is $f=sg$.

At order $ϵ$, the critical eigenfunction equation gives ${\mathcal{L}}_{R_c}(g{e}^{i{k}_{c}x})=0$. At order ${ϵ}^2$, the weakly nonlinear expansion contains the imposed second harmonic and the quadratic products of the critical mode: a horizontally uniform temperature correction proportional to $|A{|}^2$ and, in a general vertical-mode calculation, a second harmonic proportional to $A^2$. These corrections are found by solving the noncritical boundary value problems, with homogeneous thermal data except for the imposed forcing.

At order ${ϵ}^3$, the method of multiple scales produces the slow derivative $A_{\tau }$, the detuning term $R_2{\partial }_R{\mathcal{L}}_R$, and the two cross-advection terms involving first- and second-order fields. Project the $e^{i{k}_{c}x}$ component onto the adjoint eigenfunction using the vertical inner product. This is the solvability condition in the method of multiple scales: divide each resonant projection by $⟨g,g{⟩}_z$. The detuning supplies $\mu A$ with $\mu \propto R_2$; interactions of horizontal wavenumbers $2k_c$ and $-k_c$ permit $\beta A^{\ast }$ with $\beta \propto \gamma$; self-interaction through the slaved mean and second harmonic supplies $-\lambda |A{|}^{2}A$. Other products have the wrong horizontal wavenumber. Reflection $x\mapsto -x$ permits real coefficients with this cosine forcing. Thus the symmetry-allowed spatially forced convection amplitude equation isThere is a useful specialization that should not be silently missed. For the literal one-vertical-mode Stokes flow problem, the vanishing two-to-one forcing coefficient for Stokes convection makes $\beta =0$ at this order. To see this, write a positive second-harmonic forcing component as $(w,\theta )=(W(z),\Theta (z))e^{2i{k}_{c}x}$, incorporating the cosine's factor $1/2$. Its coupling to the negative critical harmonic has projected integrand, apart from sign and its factor $A^{\ast }$,The integral vanishes because $g=0$ at both plates, even though $\Theta (0)$ is nonzero. This proves the cancellation without solving the forced profiles. The permitted coefficient is therefore zero times $\gamma$; symmetry alone does not establish nonzero phase pinning for the equations actually supplied.

The same normalization makes the remaining coefficients explicit. Since ${\mathcal{W}}_{R}g=(R{k}_c^2/s^2)g$, $\mu =R_2{k}_c^2/s^2$. The quadratic second harmonic cancels for $f=sg$, while the uniform correction is ${\theta }_{20}=-s|A{|}^2\sin (2\pi z)/(2\pi )$. Projecting $-w_1{\partial }_z{\theta }_{20}$ gives $-s^2|A{|}^{2}A/2$. Thus for the literal model and this temperature normalization,A generic nonzero $\beta$ would require a nonvanishing projection in an amended physical model or a different forcing structure. It is still meaningful to classify the real-coefficient amplitude equation requested independently.

Write $A=x+iy$. Then $x_{\tau }=(\mu +\beta -\lambda (x^2+y^2))x$ and $y_{\tau }=(\mu -\beta -\lambda (x^2+y^2))y$. These are a gradient flow for $V=-\frac{1}{2}(\mu +\beta )x^2-\frac{1}{2}(\mu -\beta )y^2+\frac{\lambda }{4}(x^2+y^2{)}^2$, so local minima give stable equilibrium points. At the origin the two eigenvalues are $\mu +\beta$ and $\mu -\beta$. The origin has exponential asymptotic stability if $\mu <-|\beta |$, retains asymptotic stability with algebraic decay at $\mu =-|\beta |$, and is unstable if $\mu >-|\beta |$. At equality, ${\rho }^2=x^2+y^2$ obeys $({\rho }^2{)}_{\tau }\le -2\lambda {\rho }^4$, since both linear coefficients are nonpositive. Integrating this inequality proves attraction even in the zero-eigenvalue direction.

For $\beta >0$ the stable nonzero equilibrium points are real; for $\beta <0$ they are imaginary:The real branch has Jacobian matrix eigenvalues $-2(\mu +\beta ),-2\beta$; the imaginary branch has $-2(\mu -\beta ),2\beta$. The oppositely aligned branch, when it exists, is a saddle equilibrium. No mixed real-imaginary nonzero equilibrium is possible when $\beta \ne 0$.

For $\beta =0$, the origin is stable for $\mu \le 0$, with algebraic decay at zero. If $\mu >0$, the circle $|A{|}^2=\mu /\lambda$ is radially attracting. Each point has Lyapunov stability but has a neutral phase direction, so it does not have individual asymptotic stability; the circle has orbital stability. This is the literal model's unpinned family. The general nonzero-$\beta$ branches instead exhibit phase locking to one of two phases separated by $\pi$.

Consider a horizontal fluid layer under the Boussinesq approximation heated from below, rotating uniformly about its vertical axis. To make the linear formulas definite, take depth one, stress-free impermeable fixed-temperature plates, a horizontally infinite or sufficiently large periodic domain, and negligible centrifugal-buoyancy modifications. Rigid plates change the vertical eigenfunctions and thresholds and require their own boundary value problem; the explicit formulas below use the free-slip model. Let $P=\nu /{\kappa }_T$ be the Prandtl number, $\mathrm{Ta}=(2\Omega d^2/\nu {)}^2$ the Taylor number, and $R$ the Rayleigh number. Time is again measured on the thermal diffusion time.

The linearized rotating Rayleigh-Bénard convection equations about conduction areThe Coriolis force does no work on the fluid, but it couples vertical motion to vertical vorticity and thereby changes the balance of buoyancy and viscous damping. On the plates require $w=\theta =0$ and ${\partial }_z{u}_x={\partial }_z{u}_y=0$. Write $\zeta =(\nabla \times \mathbf{u}{)}_z$. Taking a curl and a double curl removes pressure and givesA normal mode with horizontal wavenumber $k>0$, vertical index $n\ge 1$ and complex growth rate $s$ has $w=W\sin (n\pi z)e^{st+i\mathbf{k}\cdot \mathbf{x}}$, $\theta =\Theta \sin (n\pi z)e^{st+i\mathbf{k}\cdot \mathbf{x}}$ and $\zeta =Z\cos (n\pi z)e^{st+i\mathbf{k}\cdot \mathbf{x}}$. Put $a=k^2+n^2{\pi }^2$. The three equations become $a(s/P+a)W+\sqrt{\mathrm{Ta}}n\pi Z=R{k}^2\Theta$, $(s/P+a)Z=\sqrt{\mathrm{Ta}}n\pi W$, and $(s+a)\Theta =W$. Their determinant gives the rotating-convection growth-rate polynomialThis cubic includes viscous and thermal modes as well as the convective instability. It avoids excluding a root by division by $s/P+a$. Linear stability means every root has negative real part; stationary onset has $s=0$, while overstability has $s=\pm i\omega$ with nonzero angular frequency.

Setting $s=0$ gives the stationary neutral curve of rotating convectionFor oscillatory convection, set $s=i\omega$ and equate real and imaginary parts. Eliminating $R$ yieldsThe oscillatory neutral curve of rotating convection is physically admissible only when ${\omega }^2>0$. Thus $P<1$ and sufficiently rapid rotation are necessary; a formal minimum with ${\omega }^2\le 0$ is not a Hopf bifurcation. Convection begins at the smaller of the stationary minimum and the admissible oscillatory minimum. The angular frequency approaches zero where a fixed-wave-number stationary and oscillatory threshold meet; that is a double-zero limit of the cubic, requiring a different slow-time reduction.

For wavenumber selection in rotating convection, the lowest vertical index is $n=1$. To compare indices, set $k^2=n^2{\pi }^{2}y$ at fixed $y>0$. Both neutral thresholds have a positive term proportional to $n^4$ and a rotation term independent of $n$, while the admissible angular frequency squared decreases with $n$. Their minima therefore cannot improve on $n=1$. Set $t=k^2$, $m={\pi }^2$. Minimizing each neutral curve givesFor the latter equation also check ${\omega }^2>0$, or minimize over the admissible set instead. Without rotation, $k_c=\pi /\sqrt{2}$ and $R_c=27{\pi }^4/4$. At rapid rotation, $k_s\sim (\mathrm{Ta}{\pi }^2/2{)}^{1/6}$ and $R_{s,c}\sim 3(\mathrm{Ta}{\pi }^2/2{)}^{2/3}$: rotation narrows the horizontal convection rolls and raises their threshold. In a finite box only discrete wavenumbers are permitted, so the minimum must be taken over those allowed modes.

The rapid-rotation comparison has $R_{o,c}/R_{s,c}\sim 2P^{4/3}/(1+P{)}^{1/3}$. Its equality gives $8P^4-P-1=0$, with positive root about $0.67660$. Below this value, sufficiently rapid rotation can make oscillatory convection the primary instability; low Prandtl number allows inertial motions to interact with the thermal field before viscosity damps them. This estimate describes the selected neutral-curve comparison in the stated free-slip asymptotic model, rather than claiming every $P<1$ has oscillatory primary onset at every rotation rate.

Linear selection does not determine the nonlinear planform or its stability. A weakly nonlinear expansion with a solvability condition gives amplitude equations. A stationary convection roll amplitude has the Landau amplitude equation $A_T=rA-g|A{|}^{2}A$; a supercritical branch requires $g>0$ and has $|A{|}^2=r/g$, whereas $g<0$ requires higher-order saturation and permits subcritical behaviour. The coefficient depends on the physical parameters and boundary conditions, and should not be assumed positive merely because the linear threshold is known.

A nonzero-frequency Hopf bifurcation supplies oppositely travelling convection roll amplitudes $Z_{+},Z_{-}$. At a fixed orientation, a cubic normal form iswith complex $g,h$ and the understood symmetry exchanging propagation directions. For $r>0$ and $g_r=\mathrm{Re}g>0$, a travelling-roll branch has one nonzero amplitude, of squared modulus $r/g_r$; it is stable to the opposite travelling amplitude when $h_r=\mathrm{Re}h>g_r$. A standing-roll branch has equal squared moduli $r/(g_r+h_r)$, and is stable within this pair when $g_r>h_r$ and $g_r+h_r>0$. The imaginary parts shift angular frequencies. These conditions concern amplitude perturbations in that reduced subspace; orientation and modulation modes still have to be tested. This is the competition of travelling and standing convection rolls.

Near simultaneous stationary and oscillatory thresholds, a codimension-two bifurcation requires retaining both types of mode. After selecting an orientation and fixing a steady spatial phase, a schematic nonresonant steady–Hopf mode interaction isHere $a$ is real, $z$ is the chosen complex oscillatory amplitude, $g_s,h_s$ are real, and $g_o,h_o$ may be complex. Assume $r_s>0$, $g_s>0$ for a radially stable pure steady branch, and $r_o>0$, $g_{o,r}=\mathrm{Re}(g_o)>0$ for a radially stable pure oscillatory branch. Steady convection rolls suppress the oscillatory mode if $r_o-h_{o,r}{r}_s/g_s<0$, where $h_{o,r}=\mathrm{Re}(h_o)$; oscillatory convection rolls suppress the steady mode if $r_s-h_s{r}_o/g_{o,r}<0$. These are transverse tests for branches whose existence and radial stability have already been checked. Mixed states have positive intensities $u=a^2$, $v=|z{|}^2$ solving $g_{s}u+h_{s}v=r_s$, $h_{o,r}u+g_{o,r}v=r_o$. If $D=g_s{g}_{o,r}-h_s{h}_{o,r}\ne 0$, then $u=(r_s{g}_{o,r}-h_s{r}_o)/D$, $v=(g_s{r}_o-h_{o,r}{r}_s)/D$. For positive $u,v$ and the positive self-saturation coefficients above, the intensity Jacobian matrix has negative trace and determinant $4uvD$, so the mixed state is attracting in intensities when $D>0$ and is a saddle equilibrium when $D<0$; phase and other-mode perturbations remain separate tests. Depending on the cross-couplings, one obtains coexistence or competition and bistability; degeneracies or resonances require extra terms. A full travelling/standing-wave competition must retain both $Z_{\pm }$, not just this illustrative one-mode $z$. If the Hopf angular frequency tends to zero, averaging over fast oscillations fails and a double-zero normal form must keep the corresponding two slow variables. A crossing of global minima at distinct wavenumbers is instead an interaction of distinct modes, not automatically the same double-zero problem.

Weakly nonlinear convection rolls can also lose stability spatially. The real Ginzburg–Landau equation $A_T=rA+\xi A_{XX}-g|A{|}^{2}A$, with $r,\xi ,g>0$, has detuned convection rolls with $|A{|}^2=(r-\xi Q^2)/g$. Linear phase modulation gives diffusion coefficient $\xi (r-3\xi Q^2)/(r-\xi Q^2)$, so the Eckhaus instability excludes $Q^2>r/(3\xi )$ even though convection rolls exist up to $Q^2<r/\xi$. Transverse bending and mean-flow couplings impose additional restrictions in the actual rotating layer; the scalar equation is a local longitudinal example, not its complete stability theory.

Most distinctively, rotation breaks mirror symmetry between competing convection roll orientations. For two sets of convection rolls at relative angle $\vartheta$, write ${\dot{A}}_2=r{A}_2-g_0|A_2{|}^2{A}_2-g(\vartheta )|A_1{|}^2{A}_2$ and interchange indices with $\vartheta \mapsto -\vartheta$ for the other equation. The established convection roll $|A_1{|}^2=r/g_0$ is unstable to the new orientation when its linear growth rate $r[1-g(\vartheta )/g_0]$ is positive. The inequality need not be symmetric under $\vartheta \mapsto -\vartheta$ because the imposed rotation supplies handedness. This is the Küppers–Lortz instability: sufficiently rapid rotation can destabilize steady convection rolls immediately above their stationary onset to oblique convection rolls of another orientation. Successive replacements can produce time-dependent convection roll switching or heteroclinic cycles, instead of a stable single convection roll pattern. It requires three-dimensional perturbations even when the original convection roll solution is independent of its axial coordinate.

At finite Prandtl number and stress-free plates, nearly parallel convection rolls can couple resonantly to a slowly damped large-scale Eulerian mean flow. This small-angle instability of rotating convection rolls makes a regular two-roll expansion nonuniform as the angle tends to zero; one must retain the mean-flow mode. It should not be identified automatically with the finite-angle Küppers–Lortz instability or assigned a universal threshold from a one-amplitude equation. Consequently the onset type and selected scale follow from the admissible neutral curves, while persistent convection roll patterns require a separate nonlinear and sideband stability calculation.

The full perturbation equations arewhere the streamfunction advection bracket is $J(f,g)=f_z{g}_x-f_x{g}_z$. Put $R=R_c+{ϵ}^2$ and introduce the slow time $T={ϵ}^{2}t$, so ${\partial }_t={ϵ}^2{\partial }_T$ on the amplitude-dependent fields. ThenThe weakly nonlinear expansion balances the small linear growth against the cubic saturation on this slow time scale.

The linear dispersion relation for a mode $e^{i(kx-\omega t)}$ is $5{\omega }^2=k^4+4$. The proposed modes share phase velocity $c$, so $5c^2{k}_j^2=k_j^4+4$. A quadratic wave interaction generates the second harmonic and the difference harmonic. With only two positive wavenumbers, phase matching for a quadratic wave interaction requires $k_2=2k_1$. Equating their phase velocities givesand thereforeThis is a two-to-one resonance of dispersive waves. Its amplitude changes accumulate on $t=O({ϵ}^{-1})$. Choose the slow time $T=ϵt/5$; this harmless constant rescaling makes the later amplitude formulas simple. With $\xi =x-t$, the leading real field is $A_0{e}^{i\xi }+B_0{e}^{2i\xi }+{\overline{A}}_0{e}^{-i\xi }+{\overline{B}}_0{e}^{-2i\xi }$.

For the fundamental, the order-$ϵ$ cross derivative in $5{\psi }_{tt}$ is $-2i(A_0{)}_T{e}^{i\xi }$, while the fundamental coefficient in ${\psi }_0{\psi }_{0x}$ is $i{\overline{A}}_0{B}_0$. For the second harmonic the corresponding terms are $-4i(B_0{)}_T{e}^{2i\xi }$ and $i{A}_0^2$. The solvability condition in the method of multiple scales removes the resonant forcing and givesThe nonresonant third and fourth harmonics enter the correction. They do not change these leading amplitude equations.

Write $A_0=R{e}^{i\theta }$, $B_0=P{e}^{i\varphi }$ and $\delta =\varphi -2\theta$. Taking real and imaginary parts gives the explosive two-to-one amplitude system in polar form:The polar phases are used where their corresponding amplitudes are nonzero. These equations implyFor the second identity, differentiate using ${\delta }_T=(R^2/(4P)+P)\sin \delta$: the terms proportional to $\sin \delta \cos \delta$ cancel. Thereforeare first integrals, and in particular

For constant phases with $\varphi =2\theta +\pi$, these real equations reduce to $R_T=RP/2$ and $P_T=R^2/4$. The initial data give $R^2-2P^2=8$, so the Riccati equation for $P$ is $2P_T=P^2+4$. Integrating and using $P(0)=2$ givesThe tangent and secant function have a pole at $T=\pi /4$, corresponding to $t=5\pi /(4ϵ)$. Both modes grow through their locked resonant interaction. This is formal finite-time blowup of the reduced amplitude equations; the weakly nonlinear expansion loses validity as the amplitudes become large, so it cannot establish a singularity of the full partial differential equation.

Interpret the displayed two-harmonic form as a leading term in a weakly nonlinear expansion. Write $\xi =(1+ϵ\kappa )x-t$ and ${\Psi }_0=\alpha \cos \xi +\beta \cos 2\xi$. The linear operator multiplies $\cos m\xi$ byFor $m=1,2$, this is $4ϵ\kappa +O({ϵ}^2)$ and $64ϵ\kappa +O({ϵ}^2)$, respectively. Thus detuning of a wave resonance enters at the same order as the quadratic forcing. The cosine addition formula givesProjection onto the two resonant harmonics by harmonic balance requiresThe nonzero branches are thereforeThe two signs are related by a half-period translation of $\xi$; the trivial branch $\alpha =\beta =0$ also exists.

The nonresonant first correction supplies the generated mean, third harmonic and fourth harmonic. Their linear multipliers at $ϵ=0$ are $4,40,180$, giving the nonresonant partFurther resonant amplitude corrections are determined at higher order. Consequently the nonzero answer describes an asymptotic periodic travelling wave with additional harmonics. Literally retaining only the two displayed harmonics cannot be an exact nonzero solution: their square has a positive constant term, while the linear operator applied to the two cosines has no constant term. The distinction is essential to interpreting this perturbative ansatz.

In a weakly nonlinear expansion of the long-wave convection equation with broken Boussinesq symmetry at $\mu =2$, a critical Fourier mode ${\Theta }_1=A{e}^{ix}+c.c.$ generates ${\Theta }_2=(2is/9)A^2{e}^{2ix}+c.c.$. The quadratic interaction of these first and second harmonics feeds back $8s^2|A{|}^{2}A/9$ into the critical Fourier mode, while the cubic gradient nonlinearity contributes $-3|A{|}^{2}A$. The Fredholm solvability condition is consequently ${\mu }_{2}A-g|A{|}^{2}A=0$. The sign of $g$ distinguishes supercritical and subcritical branches; $g=0$ requires a higher-order amplitude equation.

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.