Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 77 / Section I

Use the layer depth $d$, the thermal diffusion time $d^2/\kappa$, velocity $\kappa /d$, and the imposed temperature difference $\Delta T$ as scales. Let $\theta$ be the temperature departure from the conductive profile, $\mathbf{u}$ the velocity, $w=\mathbf{u}\cdot \hat{\mathbf{z}}$, and let pressure absorb hydrostatic terms. The dimensionless Boussinesq equations for Rayleigh-Bénard convection areThe Prandtl number and Rayleigh number are $\sigma =\nu /\kappa$ and $R=g{\alpha }_T\Delta T{d}^3/(\nu \kappa )$, with kinematic viscosity $\nu$, thermal diffusivity $\kappa$ and coefficient of thermal expansion ${\alpha }_T$. At $z=0,1$, the stress-free boundary condition and perfectly conducting thermal boundary condition give $w=0$, ${\partial }_z{u}_x={\partial }_z{u}_y=0$, and $\theta =0$.

For the fundamental vertical normal mode, take $w=W\sin (\pi z)e^{i\mathbf{k}\cdot \mathbf{x}+\lambda t}$ and $\theta =\vartheta \sin (\pi z)e^{i\mathbf{k}\cdot \mathbf{x}+\lambda t}$. Here $k=|\mathbf{k}|>0$ is the horizontal wavenumber. With $s=k^2+{\pi }^2$, projection onto incompressible flow eliminates pressure and givesThus the stress-free convection growth-rate polynomial is $(\lambda +\sigma s)(\lambda +s)=\sigma R{k}^2/s$.

For the infinite-Prandtl number limit, the finite thermal growth rate isThe other, viscously damped root tends to $-\sigma s$. For $\sigma =1$, both roots are explicit:The plus root determines instability. A higher vertical harmonic replaces ${\pi }^2$ by $n^2{\pi }^2$; the fundamental is the most favorable one.

For the large-$R$ maximum in the infinite-Prandtl number limit, put $x=k^2$ and $p={\pi }^2$. Differentiating gives $d{\lambda }_{\infty }/dx=R(p-x)/(p+x{)}^3-1$. The unique maximum for sufficiently large $R$ therefore satisfies $R(p-x)=(p+x{)}^3$, whenceThis is fastest growth in infinite-Prandtl convection. The large-$R$ expansion is taken after the infinite-Prandtl number limit; at finite $\sigma$, neglecting inertia additionally requires $|\lambda |\ll \sigma s$. It selects the fastest growing mode far above onset, rather than the neutral-curve minimum $k^2={\pi }^2/2$.

The Lorenz model of thermosolutal convection is a Galerkin method truncation retaining a roll velocity and the leading temperature and concentration modes. The variable $a$ measures the roll velocity; $b$ is the temperature mode correlated with the vertical motion, while $c$ is the horizontally averaged temperature-profile distortion generated by that motion. The corresponding concentration modes are $d$ and $e$. The symbol $d$ in this model is a solute amplitude, not the dimensional layer depth. The terms $ab$ and $ad$ change the mean scalar profiles; $a(1-c)$ and $a(1-e)$ express advection of the remaining background gradients.

The positive parameters are the Prandtl number $\sigma =\nu /\kappa$, the ratio $\tau ={\kappa }_s/\kappa$ of solutal diffusivity to thermal diffusivity, and the geometric vertical-mode decay ratio $\varpi$. In the usual fundamental-roll truncation with time normalized to the thermal roll-mode decay rate, $\varpi =4{\pi }^2/(k^2+{\pi }^2)$; treating it as an independently specified positive truncation parameter leaves the arguments below unchanged.

The parameter $r$ is the destabilizing thermal Rayleigh number divided by the free-slip critical-mode value, and $r_s>0$ measures stabilizing solutal buoyancy in the same thermal-diffusive normalization. For example, using a positive bottom-to-top stabilizing solute difference $\Delta S$ and density coefficient ${\beta }_S$,If a solutal Rayleigh number is instead defined with ${\kappa }_s$ in its denominator, the coefficient $r_s$ here is $\tau$ times that normalized number. Thermal buoyancy drives the roll; solutal buoyancy opposes it and diffuses at a different rate. This is the essential mechanism of thermosolutal convection.

Set the time derivatives to zero. The scalar equations giveFor the nonzero convection branch, division of the velocity equation by $a$ givesIts zero-amplitude limit is $r_e=1+r_s/\tau$. Differentiation yieldsFor $0<\tau <1$, the sharp condition for a subcritical bifurcation at the steady onset is $r_s>{\tau }^3/(1-{\tau }^2)$. The stronger bound printed in the question is sufficient, since $(1-\tau {)}^2<1-{\tau }^2$. Under it the branch initially runs toward smaller $r$, and the unique saddle-node bifurcation hasIt is a genuine positive-amplitude minimum: $r^{\prime \prime }(x)=2r_s\tau (1-{\tau }^2)/({\tau }^2+x{)}^3>0$. Put $h=\sqrt{r_s\tau }$ and $j=\sqrt{1-{\tau }^2}$. At the minimum, $1+x_{\ast }=j(j+h)$ and ${\tau }^2+x_{\ast }=hj$, soThese results describe the steady fold of a thermosolutal Lorenz model. They do not by themselves assert stability against every oscillatory disturbance.

Take $\beta \ge 0$ without loss of generality, since its definition fixes only ${\beta }^2$. With $r_s={\beta }^2\tau$, the steady-branch minimum expands as $r_{\min }=1+2\beta \tau +O({\tau }^2)$. For fixed positive $\sigma$, the source's supplied approximation therefore givesThus the requested inequality follows from the supplied approximation, with equality at first order when $\beta =\delta$. If negative $\beta$ is retained, replace it by $|\beta |$ throughout.

There is an independent source-consistency issue with calling that supplied expression the oscillatory threshold of the displayed model. Linearizing about conduction, the $c,e$ modes decay separately and the $a,b,d$ characteristic polynomial isWriting it as ${\lambda }^3+A{\lambda }^2+B\lambda +C$, a nonzero imaginary pair requires $C=AB$ and $B>0$, by the Routh-Hurwitz stability criterion. Direct algebra gives the oscillatory threshold of a thermosolutal Lorenz modelConsequently, with $\delta =\sigma /(1+\sigma )$,The two factors of $\delta$ in the supplied threshold are interchanged. For example $\sigma =1$, $\beta =2$, $\tau =10^{-3}$ gives a genuine marginal imaginary pair at $r_H=1.004003$, whereas the supplied first-order expression gives $1.0085$. The intended comparison remains true after correction:This comparison applies to a physical Hopf bifurcation when ${\omega }_H^2>0$ and to an interior steady fold when $a_{\ast }^2>0$. For fixed nonzero $\beta$ and fixed $\sigma >0$, both conditions hold for sufficiently small positive $\tau$. At a vanishing squared difference, neglected terms must be examined before drawing a higher-order conclusion.