Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 337 / 2

Assume $R^2=1-q^2>0$. The symbol $R$ in this part is the convection roll amplitude, not the Rayleigh number of the previous part. Set $r=R^2$ for the algebra. The periodic boundary condition requires $qL\in 2\pi \mathbb{Z}$ and modulation wavenumbers $lL\in 2\pi \mathbb{Z}$. The conserved-field equation preserves the imposed zero mean of $B$, since the integral of an $X$ derivative over a period vanishes.

Equating the $e^{ilX+\lambda T}$ coefficients after linearization of the convection amplitude coupled to a conserved field givesThe conjugate sideband is needed because the variation of $|A{|}^{2}A$ couples a perturbation to its complex conjugate. Put $S=a_1+a_2$, $D=a_1-a_2$. The linear operator on $(S,D,b)$ isIts characteristic polynomial, expressed conveniently without expanding every coefficient, isAt $l=0$ it has roots $-2r,0,0$. The phase root is neutral; the uniform $B$ root is excluded by the prescribed zero-flux-perturbation mean. Nonzero arbitrarily long-wave modes remain allowed on a sufficiently large domain. For fixed $r>0$, the two roots tending to zero have $\lambda =l^2\nu +O(l^4)$ at simple limiting roots. Dividing the polynomial by $2l^4$ yieldsRepeated limiting roots may require a further expansion, but do not affect the strict instability criterion. If $(\mu -\sigma )r+2\sigma q^2>0$, the constant term is negative and the two real roots have opposite signs. The positive root gives a growing long-wave sideband instability. Using $r=1-q^2$ givesThis is sufficient for instability when a sufficiently small nonzero allowed $l$ is available. It is not an unconditional instability theorem for every fixed finite period. For example $q=0$, $\mu =2$, $\sigma =1$, $L=1$ satisfy the inequality, but the nonzero allowed $l$ obey $l^2\ge 4{\pi }^2$. Their roots are $-l^2$ and $-l^2-1\pm \sqrt{1+4l^2}$, all negative. The prescribed zero mean removes the uniform flux root and the remaining uniform phase is merely neutral. The corrected conclusion is the long-wave instability with a conserved mean field, with domain size and mode availability stated.

For a real steady amplitude, integrate the conserved-field equation twice. Periodicity forces the integration slope to vanish, and the zero mean fixes the remaining constant. HenceSubstituting this into the steady amplitude equation gives the nonlocal ordinary differential equationFor $A=R\mathrm{sech}(cX)$, use the hyperbolic secant identity $A^{\prime \prime }=c^{2}A-2c^2{A}^3/R^2$. Vanishing of the two independent coefficients givesChoose $c>0$ and $R>0$; an overall negative sign is another solution. Thus $\kappa >1$. The exact average of the proposed profile over the finite interval isFor $cL\gg 1$, the hyperbolic tangent differs from one by $O(e^{-cL})$, so $cL⟨A^2⟩\simeq 2R^2$. Eliminating $R^2=2c^2/(\kappa -1)$ gives the requested leading relationEquivalently $c+c^{-1}=K$, with $K=4\kappa /[L(\kappa -1)]$. Positive real roots exist precisely when $K\ge 2$; they are $(K\pm \sqrt{K^2-4})/2$ and are reciprocal. Each candidate must additionally satisfy $cL\gg 1$. For $L>2$, the leading necessary range is $1<\kappa \le L/(L-2)$, with equality at the coalescence $c=1$. Away from this leading-order fold, the inequality is strict.

The strict finite-window condition can also be seen by retaining the average exactly. Its consistency relation is $c+c^{-1}=K\tanh (cL/2)$. Since $\tanh (cL/2)<1$, it implies $K>2$, and henceThe strict endpoint distinction is exponentially small in the large-$cL$ approximation; the leading polynomial alone permits its double root. When $0<L\le 2$, this algebra imposes no finite upper bound on $\kappa >1$. The printed interval therefore needs $L>2$: for example $L=1$, $\kappa =1.1$ gives a large positive root $c\simeq 43.98$ of the leading equation and $cL\gg 1$, despite the printed upper endpoint being negative.

Finally, the single localized pulse of a conserved-field convection model is an approximate periodic solution. Its values agree at the interval ends, but its derivatives have opposite signs there; their magnitude is exponentially small, $O(cR{e}^{-cL/2})$. Periodic-tail corrections are needed for an exact smooth periodic boundary condition. The range derived from the truncated profile is a consistency condition, not a proof that every permitted parameter gives an exact stable periodic pulse.