Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 77 / 2 / ii

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.