Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 76 (Convection)

For the long-wave convection equation with broken Boussinesq symmetry, use a sufficiently smooth real temperature field. To make the spatial average and integrations meaningful, take a periodic pattern, or an existing long-interval average with bounded derivatives and vanishing averaged endpoint fluxes. Boundedness of the temperature alone does not guarantee all those averaging properties. Multiply the evolution equation by $\Theta$ and average. Integration by parts givesand $⟨\Theta ({\Theta }_x^j{)}_x⟩=-⟨{\Theta }_x^{j+1}⟩$. Thus the energy method yieldsThe energy square completion for long-wave convection starts fromPut $v={\Theta }_x$. The energy identity becomesSince $sv-v^2=s^2/4-(v-s/2{)}^2\le s^2/4$ pointwise,Therefore $\mu <2-s^2/4$ excludes growth of the mean-square temperature, for arbitrary amplitude within this smooth averaging class. This is a nonlinear energy-stability criterion, not a proof of pointwise monotonicity at each position. A spatially constant component instead decays through the $-\Theta$ term. The criterion is sufficient; it need not coincide with the linear instability threshold.

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.

For the rotating-frame normalization of resonant amplitude forcing, use a rotating phase, $A=B\exp [i(\tilde{\Omega }T+\delta )]$. The real $\delta$ must occur inside the factor of $i$; the printed change of variables omits that $i$ on the phase constant. The correctly phased substitution givesA negative real forcing coefficient can be made positive by changing the forcing phase. Assuming $c>0$, setAll three terms then have the same coefficient scale, givingRenaming $\mathcal{T}$ as $T$ gives the requested canonical equation. This rescaling preserves forward time. If $c<0$, a forward-time normalization instead leaves a positive cubic term, and if $c=0$ a cubic normalization is impossible. If $e=0$, the unforced Landau amplitude equation must be treated separately. Thus the printed canonical form implicitly concerns saturation in a supercritical bifurcation with nonzero forcing.

To find the threefold phase-locked equilibria, write $C=R{e}^{i\theta }$ with $R>0$; separating real and imaginary parts givesA steady state satisfiessoWriting $y=R_0^2$ and $D=1+4\mu -4{\omega }^2$, the amplitude branches areThere are two distinct positive roots for $\mu >{\omega }^2-1/4$, except at $(\mu ,\omega )=(0,0)$, where $y_{-}=0$ and only the upper root is nonzero. Indeed their sum is positive in this range and their product is ${\mu }^2+{\omega }^2$. Below the fold condition there are none. At equality, there is one positive repeated root $y={\omega }^2+1/4$, the saddle-node bifurcation limit.

For each positive amplitude, the sine and cosine determine $3{\theta }_0$ modulo $2\pi$, giving three phases separated by $2\pi /3$. Thus the source's “two states” means two amplitude branches modulo the threefold spatial symmetry. Generically there are six nonzero complex equilibria, three on each branch, not literally two.

The polar Jacobian matrix at an equilibrium isOn the lower branch, $\det J=-3y_{-}\sqrt{D}<0$, so it is a saddle equilibrium and unstable. On the upper branch, $\det J=3y_{+}\sqrt{D}>0$ andbecause existence implies $\mu >-1/4$. Therefore every upper-branch equilibrium is asymptotically stable, and every lower-branch equilibrium is a saddle equilibrium, away from the degenerate endpoints. The eigenvalues are unchanged by the smooth polar coordinate transformation at $R_0>0$. The origin, not covered by those coordinates, has eigenvalues $\mu \pm i\omega$ and is stable for $\mu <0$ and unstable for $\mu >0$.

For $\omega \ne 0$, put $\mu =\hat{\mu }{\omega }^2$, $C=\omega z$ and $\tau =\omega T$, where $z=\hat{C}$. Then $C_T={\omega }^2{z}_{\tau }$ and division by ${\omega }^2$ givesWe may take $\omega >0$ without loss of the dynamics by conjugating the original equation if necessary; otherwise $\tau$ reverses the time orientation. The scaling is singular at $\omega =0$ and is not a transformation for that exactly zero-frequency case.

The Hamiltonian limit of three-to-one forcing drops the terms proportional to $\omega$. For $z=x+iy$ the remaining real system isThe proposed first integral isIndeed $x_{\tau }=H_y/2$ and $y_{\tau }=-H_x/2$, hence $H_{\tau }=0$. The unperturbed equation is a planar Hamiltonian system, with a center equilibrium at the origin and three saddle equilibria atAll saddle equilibria have $H=1/3$. The factorizationshows that their central separatrix consists of the three sides of an equilateral triangle. Each level $0<H<1/3$ inside this triangle is a closed periodic orbit. Indeed, inside the triangle $0<r<1$ and ${\partial }_{r}H=2r(1+r\sin 3\theta )>0$. Each ray from the origin therefore meets each such level once, giving a compact simple closed contour with no equilibrium point on it. The nonzero vector field traverses this contour periodically; the period grows without bound as the separatrix is approached. This supplies an infinite family, not a claim that every level outside the central region is closed.

Restore the small radial perturbation. Its exact effect on the first integral isConsequently the continuum of Hamiltonian system orbits generally does not persist. The origin becomes a weak attracting focus for $\hat{\mu }<0$ or a repelling focus for $\hat{\mu }>0$, and the three hyperbolic saddle equilibria persist with perturbed stable manifold and unstable manifold. For a small positive $\hat{\mu }$, outward drift on very small orbits balances cubic damping on somewhat larger ones, selecting a stable limit cycle rather than an arbitrary energy level. Near the center equilibrium $H\simeq r^2$, so its leading radius is $r\simeq \sqrt{\hat{\mu }}$ when $\hat{\mu }$ is also small.

For a more general closed unperturbed orbit ${\Gamma }_h$, the averaged area criterion for perturbed Hamiltonian cycles says that persistence requires the averaged energy drift ${\oint }_{{\Gamma }_h}{H}_{\tau }d\tau$ to vanish. Since the unperturbed speed is $|\nabla H|/2$, the planar divergence theorem converts this leading drift towhere ${\mathcal{D}}_h$ is the enclosed region. Isolated zeros select candidate periodic orbits; a drift changing from positive inside to negative outside gives an attracting limit cycle. The separatrix triangle has mean $r^2=1/4$, so its leading flux changes sign at $\hat{\mu }=1/2$. This marks the leading possible heteroclinic transition, with higher-order corrections needed to locate it precisely.

As a cycle approaches the saddle equilibria, long residence times and splitting of the heteroclinic cycle become important. Orbits can instead drift inward to the equilibrium point at the origin or leave the periodic island and approach one of the stable states with phase locking of the full canonical equation. Those upper-branch threefold phase-locked equilibria have $C=O(1)$, so they lie outside the local $C=O(\omega )$ scaling. Thus the small perturbation gives energy selection, attracting or repelling oscillations, and possible switching/locking transitions; it does not preserve a conserved $H$ or an infinite family of neutral periodic solutions. This qualitative picture does not assume all global parameter values have the same attractor.

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.